Asymptotics of solutions to the k -Yamabe equation near isolated - PowerPoint PPT Presentation

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Prototype question: Suppose that u is a solution to (3) in B R \ { 0 } , can one characterize the behavior of u ( x ) as x → 0? If one drops the u > 0 assumption, then there is little one can say, even in the case of c ′ = 0. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Prototype question: Suppose that u is a solution to (3) in B R \ { 0 } , can one characterize the behavior of u ( x ) as x → 0? If one drops the u > 0 assumption, then there is little one can say, even in the case of c ′ = 0. However, if u > 0 is a solution to (3) in B R \ { 0 } with c ′ = 0, then Bocher’s theorem says that u ( x ) = a | x | 2 − n + h ( x ) when n ≥ 3, where h ( x ) is a smooth harmonic function in B R . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Prototype question: Suppose that u is a solution to (3) in B R \ { 0 } , can one characterize the behavior of u ( x ) as x → 0? If one drops the u > 0 assumption, then there is little one can say, even in the case of c ′ = 0. However, if u > 0 is a solution to (3) in B R \ { 0 } with c ′ = 0, then Bocher’s theorem says that u ( x ) = a | x | 2 − n + h ( x ) when n ≥ 3, where h ( x ) is a smooth harmonic function in B R . Note that ∆ | x | 2 − n = 0 on R n \ { 0 } . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Prototype question: Suppose that u is a solution to (3) in B R \ { 0 } , can one characterize the behavior of u ( x ) as x → 0? If one drops the u > 0 assumption, then there is little one can say, even in the case of c ′ = 0. However, if u > 0 is a solution to (3) in B R \ { 0 } with c ′ = 0, then Bocher’s theorem says that u ( x ) = a | x | 2 − n + h ( x ) when n ≥ 3, where h ( x ) is a smooth harmonic function in B R . Note that ∆ | x | 2 − n = 0 on R n \ { 0 } . The same question for the c ′ > 0 case was resolved in the celebrated work of Caffarelli, Gidas, and Spruck, and will be reviewed later. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case We often write (2) in terms of different variables. Introduce cylindrical variables t = − ln | x | and θ = x / | x | (so that | dx | 2 = | x | 2 ( dt 2 + d θ 2 ) ), Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case We often write (2) in terms of different variables. Introduce cylindrical variables t = − ln | x | and θ = x / | x | (so that | dx | 2 = | x | 2 ( dt 2 + d θ 2 ) ), and new variables U ( t , θ ) and w ( t , θ ) by n − 2 ( x ) | dx | 2 = U 4 n − 2 ( t , θ )( dt 2 + d θ 2 ) 4 u = e − 2 w ( t ,θ ) ( dt 2 + d θ 2 ) . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case We often write (2) in terms of different variables. Introduce cylindrical variables t = − ln | x | and θ = x / | x | (so that | dx | 2 = | x | 2 ( dt 2 + d θ 2 ) ), and new variables U ( t , θ ) and w ( t , θ ) by n − 2 ( x ) | dx | 2 = U 4 n − 2 ( t , θ )( dt 2 + d θ 2 ) 4 u = e − 2 w ( t ,θ ) ( dt 2 + d θ 2 ) . Thus n − 2 2 u ( x ) = U ( t , θ ) = e − n − 2 2 w ( t ,θ ) . | x | Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case We often write (2) in terms of different variables. Introduce cylindrical variables t = − ln | x | and θ = x / | x | (so that | dx | 2 = | x | 2 ( dt 2 + d θ 2 ) ), and new variables U ( t , θ ) and w ( t , θ ) by n − 2 ( x ) | dx | 2 = U 4 n − 2 ( t , θ )( dt 2 + d θ 2 ) 4 u = e − 2 w ( t ,θ ) ( dt 2 + d θ 2 ) . Thus n − 2 2 u ( x ) = U ( t , θ ) = e − n − 2 2 w ( t ,θ ) . | x | Written in terms of U ( t , θ ) and with c ′ = n ( n − 2 ) / 4, (3) becomes U tt ( t , θ )+∆ S n − 1 U ( t , θ ) − ( n − 2 ) 2 U ( t , θ )+ n ( n − 2 ) n + 2 n − 2 ( t , θ ) = 0 . U 4 4 (4) Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case When w ( t , θ ) = ξ ( t ) is a radial solution to (2), written with respect to the background metric dt 2 + d θ 2 and c normalized to be 2 − k � n � , k Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case When w ( t , θ ) = ξ ( t ) is a radial solution to (2), written with respect to the background metric dt 2 + d θ 2 and c normalized to be 2 − k � n � , (2) reduces to the following ODE k � k � n ξ tt + ( 1 2 − k e 2 k ξ = 1 . 2 ( 1 − ξ 2 t ) k − 1 n )( 1 − ξ 2 t ) (5) Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case When w ( t , θ ) = ξ ( t ) is a radial solution to (2), written with respect to the background metric dt 2 + d θ 2 and c normalized to be 2 − k � n � , (2) reduces to the following ODE k � k � n ξ tt + ( 1 2 − k e 2 k ξ = 1 . 2 ( 1 − ξ 2 t ) k − 1 n )( 1 − ξ 2 t ) (5) (5) is not fully nonlinear, but may become degenerate in the case k > 1 when 1 − ξ 2 t → 0. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case For a metric g whose Schouten-Weyl tensor is in the Γ + k class with k > 1, we know (i) σ k ( g − 1 ◦ A g ) places a stronger control on the curvature tensor than the scalar curvature: Chang, Gursky and Yang [CGY1] observed that if σ 1 ( g − 1 ◦ A g ) , σ 2 ( g − 1 ◦ A g ) > 0 at a point on a 4-dimensional manifold, then the Ricci tensor of g is positive definite at that point; Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case For a metric g whose Schouten-Weyl tensor is in the Γ + k class with k > 1, we know (i) σ k ( g − 1 ◦ A g ) places a stronger control on the curvature tensor than the scalar curvature: Chang, Gursky and Yang [CGY1] observed that if σ 1 ( g − 1 ◦ A g ) , σ 2 ( g − 1 ◦ A g ) > 0 at a point on a 4-dimensional manifold, then the Ricci tensor of g is positive definite at that point; this algebraic relation has been generalized to higher dimensions by Guan, Viaclovsky, and Wang [GVW]; and (ii) the expression σ k ( g − 1 ◦ A g ) is a fully nonlinear PDO in w that becomes elliptic . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Singular sets and the asymptotics of singular solutions to equation (2) are often intimately related to the underlying conformal geometry of the manifold. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Singular sets and the asymptotics of singular solutions to equation (2) are often intimately related to the underlying conformal geometry of the manifold. Loewner and Nirenberg studied solutions to (3) on R n \ Γ with c = − 1, where Γ stands for the singular set for u . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Singular sets and the asymptotics of singular solutions to equation (2) are often intimately related to the underlying conformal geometry of the manifold. Loewner and Nirenberg studied solutions to (3) on R n \ Γ with c = − 1, where Γ stands for the singular set for u . They showed that if Γ is a closed set with Hausdorff dimension < ( n − 2 ) / 2, then any solution u to (3) on R n \ Γ with c = − 1 is in fact smooth across Γ , Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case and that if Γ is a closed smooth submanifold of R n with dimension > ( n − 2 ) / 2, then there is a solution to (3) on R n \ Γ with c = − 1 which has non-removable singularity over Γ . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case and that if Γ is a closed smooth submanifold of R n with dimension > ( n − 2 ) / 2, then there is a solution to (3) on R n \ Γ with c = − 1 which has non-removable singularity over Γ . Later improvements and generalizations were obtained by Aviles, Veron, McOwen, D. Finn, and others. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case and that if Γ is a closed smooth submanifold of R n with dimension > ( n − 2 ) / 2, then there is a solution to (3) on R n \ Γ with c = − 1 which has non-removable singularity over Γ . Later improvements and generalizations were obtained by Aviles, Veron, McOwen, D. Finn, and others. Schoen and Yau studied solutions to (3) on R n \ Γ with c = 1 in connection with their study of developing maps of locally conformally flat manifolds. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case and that if Γ is a closed smooth submanifold of R n with dimension > ( n − 2 ) / 2, then there is a solution to (3) on R n \ Γ with c = − 1 which has non-removable singularity over Γ . Later improvements and generalizations were obtained by Aviles, Veron, McOwen, D. Finn, and others. Schoen and Yau studied solutions to (3) on R n \ Γ with c = 1 in connection with their study of developing maps of locally conformally flat manifolds. Their results imply that if u is a solution to (3) on R n \ Γ with c = 1 such that the corresponding conformal metric g is complete, then the Hausdorff dimension of Γ is ≤ ( n − 2 ) / 2. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Schoen also constructed solutions to (3) on R n \ Γ with c = 1 that are singular at Γ , where Γ is any prescribed finite set of more than one point. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Schoen also constructed solutions to (3) on R n \ Γ with c = 1 that are singular at Γ , where Γ is any prescribed finite set of more than one point. Constructions of solutions with more general prescribed singular sets, subject to the dimensional constraints, were later obtained by N. Smale, R. Mazzeo, F . Pacard, and others. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Schoen also constructed solutions to (3) on R n \ Γ with c = 1 that are singular at Γ , where Γ is any prescribed finite set of more than one point. Constructions of solutions with more general prescribed singular sets, subject to the dimensional constraints, were later obtained by N. Smale, R. Mazzeo, F . Pacard, and others. Chang, Hang, and Yang proved that if Ω ⊂ S n ( n ≥ 5) admits a complete, conformal metric g σ 1 ( A g ) ≥ c 1 > 0 , σ 2 ( A g ) ≥ 0 , and | R g | + |∇ g R | g ≤ c 0 , (6) then dim ( S n \ Ω) < ( n − 4 ) / 2. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case This has been generalized by M. Gonzalez to the case of 2 < k < n / 2: if Ω ⊂ S n admits a complete, conformal metric g with σ 1 ( A g ) ≥ c 1 > 0 , σ 2 ( A g ) , · · · , σ k ( A g ) ≥ 0, and (6), then dim ( S n \ Ω) < ( n − 2 k ) / 2. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Global solutions to (3) on R n , or on R n with one point deleted, were classified by Caffarelli, Gidas, and Spruck (earlier, related results were due to Obata, and Gidas-Ni-Nirenberg). Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Global solutions to (3) on R n , or on R n with one point deleted, were classified by Caffarelli, Gidas, and Spruck (earlier, related results were due to Obata, and Gidas-Ni-Nirenberg). Furthermore, they obtained asymptotics for solutions to (3) with isolated singularity. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Global solutions to (3) on R n , or on R n with one point deleted, were classified by Caffarelli, Gidas, and Spruck (earlier, related results were due to Obata, and Gidas-Ni-Nirenberg). Furthermore, they obtained asymptotics for solutions to (3) with isolated singularity. More precisely, they proved Theorem A Suppose that u ( x ) is a positive solution to (3) in B R \ { 0 } and does not extend to a smooth solution to (3) over 0, then u ( x ) = ¯ u ( | x | ) ( 1 + O ( | x | )) as x → 0 , (7) with � ¯ u ( | x | ) = \ u ( | x | θ ) d θ Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated being the spherical average of u over the sphere ∂ B ( 0 ) ;

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case furthermore, there exists a radial singular solution u ∗ ( | x | ) to (3) on R n \ { 0 } and some α > 0, 0 < ǫ ≤ ǫ 0 and τ such that u ( x ) = u ∗ ( | x | ) ( 1 + O ( | x | α )) as | x | → 0. (8) and u ∗ ( | x | ) = | x | − n − 2 2 ψ ǫ ( − ln | x | + τ ) as | x | → 0, where ψ ǫ ( t ) is an entire radial solution to (4) on R . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case furthermore, there exists a radial singular solution u ∗ ( | x | ) to (3) on R n \ { 0 } and some α > 0, 0 < ǫ ≤ ǫ 0 and τ such that u ( x ) = u ∗ ( | x | ) ( 1 + O ( | x | α )) as | x | → 0. (8) and u ∗ ( | x | ) = | x | − n − 2 2 ψ ǫ ( − ln | x | + τ ) as | x | → 0, where ψ ǫ ( t ) is an entire radial solution to (4) on R . Alternative approach and refined asymptotics for the same problem were later obtained by Korevaar, Mazzeo, Pacard and Schoen. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case furthermore, there exists a radial singular solution u ∗ ( | x | ) to (3) on R n \ { 0 } and some α > 0, 0 < ǫ ≤ ǫ 0 and τ such that u ( x ) = u ∗ ( | x | ) ( 1 + O ( | x | α )) as | x | → 0. (8) and u ∗ ( | x | ) = | x | − n − 2 2 ψ ǫ ( − ln | x | + τ ) as | x | → 0, where ψ ǫ ( t ) is an entire radial solution to (4) on R . Alternative approach and refined asymptotics for the same problem were later obtained by Korevaar, Mazzeo, Pacard and Schoen. There have also been many other papers related to the themes of Theorem A, including a recent work of Marques dealing with background metric not necessarily locally conformally flat in low dimensions. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Our work is to extend these asymptotics results to (admissible) solutions of (2) with isolated singularities. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Our work is to extend these asymptotics results to (admissible) solutions of (2) with isolated singularities. Other related questions are: Given a subset S of S n , contruct (admissible) solutions of (2) on S n \ S which are singulalr on S and gives rise to a complete metric on S n \ S . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Our work is to extend these asymptotics results to (admissible) solutions of (2) with isolated singularities. Other related questions are: Given a subset S of S n , contruct (admissible) solutions of (2) on S n \ S which are singulalr on S and gives rise to a complete metric on S n \ S . Given a subset S of S n , study the moduli space of (admissible) solutions of (2) on S n \ S which are singulalr on S and gives rise to a complete metric on S n \ S . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

A few relevant geometric notions Description of the problem Various forms of the σ k -Yamabe equation Why study solutions in Γ + Statements of Theorems k class? Discussion of proof Why study the asymptotics of singular solutions to equation (2)? Known results about k = 1 case Our work is to extend these asymptotics results to (admissible) solutions of (2) with isolated singularities. Other related questions are: Given a subset S of S n , contruct (admissible) solutions of (2) on S n \ S which are singulalr on S and gives rise to a complete metric on S n \ S . Given a subset S of S n , study the moduli space of (admissible) solutions of (2) on S n \ S which are singulalr on S and gives rise to a complete metric on S n \ S . L. Mazzieri, C. B. Ndiaye and A. Segatti have obtained some results to the first question above. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion First, we record part of the results by Chang, Han and Yang on the classification of global radial solutions to (2) that will be useful in our settings. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion First, we record part of the results by Chang, Han and Yang on the classification of global radial solutions to (2) that will be useful in our settings. In very rough terms, w ( t ) = ln cosh ( t ) is always a solution to (2), which corresponds to u 4 / ( n − 2 ) ( x ) | dx | 2 being the round spherical metric � � 2 2 | dx | 2 . 1 + | x | 2 Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion First, we record part of the results by Chang, Han and Yang on the classification of global radial solutions to (2) that will be useful in our settings. In very rough terms, w ( t ) = ln cosh ( t ) is always a solution to (2), which corresponds to u 4 / ( n − 2 ) ( x ) | dx | 2 being the round spherical metric � � 2 2 | dx | 2 . 1 + | x | 2 when 2 k < n , there exists a two parameter family of undulating solutions (called Delaunay type solutions in other contexts), periodic in t . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion When 2 k > n , all radial solutions defined in B R \ { 0 } have the form u ( x ) = u 0 + A | x | 2 − n / k + h . o . t . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion When 2 k > n , all radial solutions defined in B R \ { 0 } have the form u ( x ) = u 0 + A | x | 2 − n / k + h . o . t . When 2 k = n , all radial solutions defined in B R \ { 0 } , in terms of U , have the form � √ � √ 1 − k − n − 2 h t 2 U = U 0 e + h . o . t , for some 0 ≤ h < 1, as | x | = e − t → 0. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion For any solution ξ ( t ) of (5), t ( t )) k − e − n ξ ( t ) e ( 2 k − n ) ξ ( t ) ( 1 − ξ 2 is a constant, which we call the first integral of the solution ξ ( t ) . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion For any solution ξ ( t ) of (5), t ( t )) k − e − n ξ ( t ) e ( 2 k − n ) ξ ( t ) ( 1 − ξ 2 is a constant, which we call the first integral of the solution ξ ( t ) . We can parametrize the global singular radial solutions to (5) as follows: for each 0 < h , subject to any further constraints depending on 2 k < or = n , let ξ h ( t ) denote the solution to (5) with its first integral equal to h and such that ξ h ( 0 ) equals min R ξ h ( t ) . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion For any solution ξ ( t ) of (5), t ( t )) k − e − n ξ ( t ) e ( 2 k − n ) ξ ( t ) ( 1 − ξ 2 is a constant, which we call the first integral of the solution ξ ( t ) . We can parametrize the global singular radial solutions to (5) as follows: for each 0 < h , subject to any further constraints depending on 2 k < or = n , let ξ h ( t ) denote the solution to (5) with its first integral equal to h and such that ξ h ( 0 ) equals min R ξ h ( t ) . Next we state our main result. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion Theorem 1 Let w ( t , θ ) be a solution to (2) on { t > t 0 } × S n − 1 in the Γ + class, where the constant is normalized to be 2 − k � n � k . Then k there exists α > 0 , h and τ such that | w ( t , θ ) − ξ h ( t + τ ) | ≤ Ce − α t . (9) Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion Theorem 1 Let w ( t , θ ) be a solution to (2) on { t > t 0 } × S n − 1 in the Γ + class, where the constant is normalized to be 2 − k � n � k . Then k there exists α > 0 , h and τ such that | w ( t , θ ) − ξ h ( t + τ ) | ≤ Ce − α t . (9) Remarks In terms of the variable u ( x ) defined on B R \ { 0 } , if n − 2 2 u ( x ) > 0 , lim inf x → 0 | x | (10) then 2 k < n and 0 < h ≤ h ∗ , furthermore Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion u ( x ) = ( 1 + o ( | x | α )) | x | − n − 2 2 e − n − 2 2 ξ h ( − ln | x | + τ ) (11) as x → 0; Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion u ( x ) = ( 1 + o ( | x | α )) | x | − n − 2 2 e − n − 2 2 ξ h ( − ln | x | + τ ) (11) as x → 0; if 2 k > n , or 2 k < n and n − 2 2 u ( x ) = 0 , lim inf x → 0 | x | (12) then lim x → 0 u ( x ) exists and equals some a > 0, and there exist some α > 0 and C > 0 such that | u ( x ) − a | ≤ C | x | α ; (13) Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion if 2 k = n , then 0 ≤ h < 1 and for some α > 0, √ √ n − 2 1 − k 2 ( 1 − h ) u ( x ) | x | extends to a C α positive function over B R . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion if 2 k = n , then 0 ≤ h < 1 and for some α > 0, √ √ n − 2 1 − k 2 ( 1 − h ) u ( x ) | x | extends to a C α positive function over B R . Remarks In the case 2 k > n Gursky and Viaclovsky, YanYan Li had obtained (13) earlier. In the case 2 k < n, M. Gonzalez proved that if u is a solution to (2) in B R \ { 0 } in the Γ + k class such that u 4 / ( n − 2 ) | dx | 2 has finite volume over B R \ { 0 } , then u is bounded in B R \ { 0 } . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion As in [KMPS], we also obtain higher order expansions for solutions to (2) in the case 2 k ≤ n . Theorem 2 Let w ( t , θ ) be a solution to (2) on { t > t 0 } × S n − 1 in the Γ + k class, where n ≥ 3 , 2 ≤ k ≤ n / 2 , and the constant c is normalized to be 2 − k � n � , k Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion As in [KMPS], we also obtain higher order expansions for solutions to (2) in the case 2 k ≤ n . Theorem 2 Let w ( t , θ ) be a solution to (2) on { t > t 0 } × S n − 1 in the Γ + k class, where n ≥ 3 , 2 ≤ k ≤ n / 2 , and the constant c is normalized to be 2 − k � n � , and let w ∗ ( t ) = ξ h ( t + τ ) be the radial k solution to (2) on R × S n − 1 in the Γ + k class for which (9) holds. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion As in [KMPS], we also obtain higher order expansions for solutions to (2) in the case 2 k ≤ n . Theorem 2 Let w ( t , θ ) be a solution to (2) on { t > t 0 } × S n − 1 in the Γ + k class, where n ≥ 3 , 2 ≤ k ≤ n / 2 , and the constant c is normalized to be 2 − k � n � , and let w ∗ ( t ) = ξ h ( t + τ ) be the radial k solution to (2) on R × S n − 1 in the Γ + k class for which (9) holds. Let { Y j ( θ ) : j = 0 , 1 , · · · } denote the set of normalized spherical harmonics, Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion As in [KMPS], we also obtain higher order expansions for solutions to (2) in the case 2 k ≤ n . Theorem 2 Let w ( t , θ ) be a solution to (2) on { t > t 0 } × S n − 1 in the Γ + k class, where n ≥ 3 , 2 ≤ k ≤ n / 2 , and the constant c is normalized to be 2 − k � n � , and let w ∗ ( t ) = ξ h ( t + τ ) be the radial k solution to (2) on R × S n − 1 in the Γ + k class for which (9) holds. Let { Y j ( θ ) : j = 0 , 1 , · · · } denote the set of normalized spherical harmonics, and ρ be the infimum of the positive characteristic exponents to the linearized equation of (2) at w ∗ ( t ) corresponding to higher order spherical harmonics Y j ( θ ) , j > n Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion Then ρ > 1, and there is a a j e − t − τ � � n � ′ w 1 ( t , θ ) = 1 + ξ h ( t + τ ) Y j ( θ ) , j = 1 which is a solution to the linearized equation of (2) at w ∗ ( t ) , Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion Then ρ > 1, and there is a a j e − t − τ � � n � ′ w 1 ( t , θ ) = 1 + ξ h ( t + τ ) Y j ( θ ) , j = 1 which is a solution to the linearized equation of (2) at w ∗ ( t ) , such that | w ( t , θ ) − w ∗ ( t ) − w 1 ( t , θ ) | ≤ Ce − min { 2 ,ρ } t for t > t 0 + 1 , (14) provided ρ � = 2; Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion Then ρ > 1, and there is a a j e − t − τ � � n � ′ w 1 ( t , θ ) = 1 + ξ h ( t + τ ) Y j ( θ ) , j = 1 which is a solution to the linearized equation of (2) at w ∗ ( t ) , such that | w ( t , θ ) − w ∗ ( t ) − w 1 ( t , θ ) | ≤ Ce − min { 2 ,ρ } t for t > t 0 + 1 , (14) provided ρ � = 2; when ρ = 2, (14) continues to hold if the right hand side is modified into Cte − 2 t . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Description of the problem Classification of global radial solutions to the σ k -Yamabe equation Statements of Theorems Main theorem Discussion of proof Higer order expansion Then ρ > 1, and there is a a j e − t − τ � � n � ′ w 1 ( t , θ ) = 1 + ξ h ( t + τ ) Y j ( θ ) , j = 1 which is a solution to the linearized equation of (2) at w ∗ ( t ) , such that | w ( t , θ ) − w ∗ ( t ) − w 1 ( t , θ ) | ≤ Ce − min { 2 ,ρ } t for t > t 0 + 1 , (14) provided ρ � = 2; when ρ = 2, (14) continues to hold if the right hand side is modified into Cte − 2 t . Theorem 2 requires some knowledge on the spectrum of the linearized operator to (2). Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE We are able to provide the needed analysis. Such analysis will also be needed in constructing solutions to (2) on S n \ Λ , and in analysing the moduli space of solutions to (2) on S n \ Λ , when Λ is a finite set. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE We are able to provide the needed analysis. Such analysis will also be needed in constructing solutions to (2) on S n \ Λ , and in analysing the moduli space of solutions to (2) on S n \ Λ , when Λ is a finite set. Our knowledge of the spectrum of the linearized operator to (2) immediately yields Fredholm mapping properties of these operators on appropriately defined weighted spaces, as those in [MPU], [MS], and [KMPS]. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE We are able to provide the needed analysis. Such analysis will also be needed in constructing solutions to (2) on S n \ Λ , and in analysing the moduli space of solutions to (2) on S n \ Λ , when Λ is a finite set. Our knowledge of the spectrum of the linearized operator to (2) immediately yields Fredholm mapping properties of these operators on appropriately defined weighted spaces, as those in [MPU], [MS], and [KMPS]. We are able to adapt either of the approach in [CGS] or [KMPS] to prove the main part of Theorem 1. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE The approach in [CGS] first proves that w ( t , θ ) is well approximated by its spherical average, then proves that its spherical average can be approximated by a singular global radial solution to (3); Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE The approach in [CGS] first proves that w ( t , θ ) is well approximated by its spherical average, then proves that its spherical average can be approximated by a singular global radial solution to (3); while the approach in [KMPS] makes use of rescaling technique to extract subsequential convergence (on compact domains) to singular global radial solutions, Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE The approach in [CGS] first proves that w ( t , θ ) is well approximated by its spherical average, then proves that its spherical average can be approximated by a singular global radial solution to (3); while the approach in [KMPS] makes use of rescaling technique to extract subsequential convergence (on compact domains) to singular global radial solutions, and makes essential use of the spectrum of the linearized operator to (2) at the singular global radial solution to (2). Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE The approach in [CGS] first proves that w ( t , θ ) is well approximated by its spherical average, then proves that its spherical average can be approximated by a singular global radial solution to (3); while the approach in [KMPS] makes use of rescaling technique to extract subsequential convergence (on compact domains) to singular global radial solutions, and makes essential use of the spectrum of the linearized operator to (2) at the singular global radial solution to (2). Both approaches need a Liouville type classification results which show that all global singular solution to (2) are radial. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE The needed classification results are obtained by combining a Liouville type result of Y.Y Li for solutions to equations that include (2) and the classification results of radial solutions to (2). Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE The needed classification results are obtained by combining a Liouville type result of Y.Y Li for solutions to equations that include (2) and the classification results of radial solutions to (2). Proposition 1 Let U ( t , θ ) = e − n − 2 2 w ( t ,θ ) be any positive solution to (2) defined on the entire cylinder 4 n − 2 ( t , θ )( dt 2 + d θ 2 ) is in the Γ + R × S n − 1 . Suppose that U k class. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE The needed classification results are obtained by combining a Liouville type result of Y.Y Li for solutions to equations that include (2) and the classification results of radial solutions to (2). Proposition 1 Let U ( t , θ ) = e − n − 2 2 w ( t ,θ ) be any positive solution to (2) defined on the entire cylinder 4 n − 2 ( t , θ )( dt 2 + d θ 2 ) is in the Γ + R × S n − 1 . Suppose that U k class.Then either 2 U ( − ln | x | , x u ( x ) = | x | − n − 2 | x | ) Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE can be extended as a C 2 positive function near 0, in which case u ( x ) is a radially symmetric solution to (2) about some point in R n ; Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE can be extended as a C 2 positive function near 0, in which case u ( x ) is a radially symmetric solution to (2) about some point in R n ; or u ( x ) can’t be extended as a C 2 positive function near 0, and U is independent of θ . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE can be extended as a C 2 positive function near 0, in which case u ( x ) is a radially symmetric solution to (2) about some point in R n ; or u ( x ) can’t be extended as a C 2 positive function near 0, and U is independent of θ . Moreover, when 2 k < n , U ( t ) is a periodic solution of (1) with 0 < U ( t ) ≤ 1 for all t ∈ R and the first integral h > 0. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE can be extended as a C 2 positive function near 0, in which case u ( x ) is a radially symmetric solution to (2) about some point in R n ; or u ( x ) can’t be extended as a C 2 positive function near 0, and U is independent of θ . Moreover, when 2 k < n , U ( t ) is a periodic solution of (1) with 0 < U ( t ) ≤ 1 for all t ∈ R and the first integral h > 0. We refer to these solutions as global singular solutions to (1). Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE Let me first outline a few key steps in our proof using the spectrum analysis for the case 2 k < n . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE Let me first outline a few key steps in our proof using the spectrum analysis for the case 2 k < n . The asymptotic behavior of u ( x ) as x → 0 is encoded in the asymptotic behavior of w ( t , θ ) as t → ∞ , which we will study through the limit(s) of { w ( t + t j , θ ) } for any sequence t j → ∞ . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE Let me first outline a few key steps in our proof using the spectrum analysis for the case 2 k < n . The asymptotic behavior of u ( x ) as x → 0 is encoded in the asymptotic behavior of w ( t , θ ) as t → ∞ , which we will study through the limit(s) of { w ( t + t j , θ ) } for any sequence t j → ∞ . Step 1. Get an L ∞ bound for w ( t , θ ) , so that the sequence { w ( t + t j , θ ) } is bounded for any sequence t j → ∞ . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE Let me first outline a few key steps in our proof using the spectrum analysis for the case 2 k < n . The asymptotic behavior of u ( x ) as x → 0 is encoded in the asymptotic behavior of w ( t , θ ) as t → ∞ , which we will study through the limit(s) of { w ( t + t j , θ ) } for any sequence t j → ∞ . Step 1. Get an L ∞ bound for w ( t , θ ) , so that the sequence { w ( t + t j , θ ) } is bounded for any sequence t j → ∞ . Once an L ∞ bound for w ( t , θ ) is obtained, we will aim for gradient estimates of w ( t , θ ) . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE Let me first outline a few key steps in our proof using the spectrum analysis for the case 2 k < n . The asymptotic behavior of u ( x ) as x → 0 is encoded in the asymptotic behavior of w ( t , θ ) as t → ∞ , which we will study through the limit(s) of { w ( t + t j , θ ) } for any sequence t j → ∞ . Step 1. Get an L ∞ bound for w ( t , θ ) , so that the sequence { w ( t + t j , θ ) } is bounded for any sequence t j → ∞ . Once an L ∞ bound for w ( t , θ ) is obtained, we will aim for gradient estimates of w ( t , θ ) . In fact, for solution w ( t , θ ) to (2), there is local gradient estimates of w ( t , θ ) in terms of an upper bound for e − w ( t ,θ ) through the work of Guan-Wang. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE An upper bound for e − w ( t ,θ ) is obtained throught the following result of Y.Y. Li. Proposition 2 Suppose that u ∈ C 2 ( B 2 \ { 0 } ) is a positive function such that 4 n − 2 ( x ) | dx | 2 g = u is a metric in Γ + k and gives rise to a solution to (1) over B 2 \ { 0 } . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE An upper bound for e − w ( t ,θ ) is obtained throught the following result of Y.Y. Li. Proposition 2 Suppose that u ∈ C 2 ( B 2 \ { 0 } ) is a positive function such that 4 n − 2 ( x ) | dx | 2 g = u is a metric in Γ + k and gives rise to a solution to (1) over B 2 \ { 0 } . Then n − 2 2 u ( x ) < ∞ , lim sup | x | (15) x → 0 Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE and there exists some constant C > 0 such that | u ( x ) − ¯ u ( | x | ) | ≤ C | x | ¯ u ( | x | ) , (16) for 0 < | x | ≤ 1, where � 1 ¯ u ( | x | ) = u ( y ) d σ ( y ) | ∂ B | x | ( 0 ) | ∂ B | x | ( 0 ) is the spherical average of u ( x ) over ∂ B | x | ( 0 ) . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE In terms of the variables 2 2 n − 2 u ( e − t θ ) + t , w ( t , θ ) = − n − 2 ln U ( t , θ ) = − and � γ ( t ) := | S n − 1 | − 1 S n − 1 w ( t , θ ) d θ, Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE In terms of the variables 2 2 n − 2 u ( e − t θ ) + t , w ( t , θ ) = − n − 2 ln U ( t , θ ) = − and � γ ( t ) := | S n − 1 | − 1 S n − 1 w ( t , θ ) d θ, (15) implies that e − 2 w ( t ,θ ) ≤ C 1 , (17) for some constant C 1 , Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE In terms of the variables 2 2 n − 2 u ( e − t θ ) + t , w ( t , θ ) = − n − 2 ln U ( t , θ ) = − and � γ ( t ) := | S n − 1 | − 1 S n − 1 w ( t , θ ) d θ, (15) implies that e − 2 w ( t ,θ ) ≤ C 1 , (17) for some constant C 1 , and (16) implies that, � w ( t , θ ) := w ( t , θ ) − γ ( t ) satisfies w ( t , θ ) | ≤ C 2 e − t , | � (18) for some constant C 2 . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE Using (18), gradients estimates and interpolation, we can obtain, for any 1 > δ > 0, a constant C > 0 such that |∇ l t ,θ ( w ( t , θ ) − γ ( t )) | ≤ Ce − ( 1 − δ ) t , (19) for all t ≥ 0 and 1 ≤ l ≤ 2. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE Using (18), gradients estimates and interpolation, we can obtain, for any 1 > δ > 0, a constant C > 0 such that |∇ l t ,θ ( w ( t , θ ) − γ ( t )) | ≤ Ce − ( 1 − δ ) t , (19) for all t ≥ 0 and 1 ≤ l ≤ 2. Step 2. Make use of a Pohozaev type identity for solutions w ( t , θ ) to (2), which takes the form � � � n � ne ( 2 k − n ) w ( t ,θ ) d θ = � T a 1 [ w ( t , θ )] w at ( t , θ ) − e − nw ( t ,θ ) h 2 k σ k S n − 1 a = 1 (20) for some constant � h independent of t , Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE Using (18), gradients estimates and interpolation, we can obtain, for any 1 > δ > 0, a constant C > 0 such that |∇ l t ,θ ( w ( t , θ ) − γ ( t )) | ≤ Ce − ( 1 − δ ) t , (19) for all t ≥ 0 and 1 ≤ l ≤ 2. Step 2. Make use of a Pohozaev type identity for solutions w ( t , θ ) to (2), which takes the form � � � n � ne ( 2 k − n ) w ( t ,θ ) d θ = � T a 1 [ w ( t , θ )] w at ( t , θ ) − e − nw ( t ,θ ) h 2 k σ k S n − 1 a = 1 (20) for some constant � h independent of t , where T a 1 [ w ( t , θ )] are the components of the Newton tensor associated with σ k ( A w ( t ,θ ) ) . Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated

Outline of key steps in proof More details on the spectrum analysis Description of the problem Spectrum analysis for higher order expansion Statements of Theorems Proof for Theorem 2 Discussion of proof Another proof using approximating ODE An asymptotic result on ODE (20) can be expressed in terms of γ ( t ) via the help of estimate (19), as e ( 2 k − n ) γ � � t ) k + η 3 ( t ) − e − n γ { 1 + η 4 ( t ) } = h , ( 1 − γ 2 (21) where h ≥ 0 is a constant multiple of � h , η i ( t ) , for i = 3 , 4, have the decay rate η i ( t ) = O ( e − 2 ( 1 − δ ) t ) as t → ∞ , and δ > 0 can be made as small as one needs. Zheng-Chao Han, YanYan Li, Eduardo Teixeira Asymptotics of solutions to the σ k -Yamabe equation near isolated