Isoparametric hypersurfaces and the Yamabe equation Guillermo Henry - PowerPoint PPT Presentation

Isoparametric hypersurfaces and the Yamabe equation Guillermo Henry Universidad de Buenos Aires- CONICET Workshop on the Isoparametric Theory 2-6 June, 2019, BNU, Beijing. Isoparametric hypersurfaces and the Yamabe equation

This talk is mainly based on a joint work with Jimmy Petean, CIMAT, Guanajuato, GTO, Mexico. Isoparametric hypersurfaces and the Yamabe equation

Let ( M n , g ) be a closed connected Riemannian manifold of dimension n ≥ 3. A function u : M − → I R is a solution of the Yamabe equation if a n ∆ g u + s g u = cu p n − 1 for some c ∈ I R , where s g is the scalar curvature of ( M , g ), a n = 4( n − 1) R n f = − � ∂ 2 f 2 n ( n − 2) , and p n = n − 2 . ( ∆ I i ) i ∂ x 2 We are interested in both positive solutions and nodal solutions ( i.e., sign changing solutions). Isoparametric hypersurfaces and the Yamabe equation

Let ( M n , g ) be a closed Riemannian manifold ( n ≥ 3) and let S ⊂ M be an isoparametric hypersurface. Does there exist a solution of the Yamabe equation (positive/ nodal) that is constant along S ? We are going to discuss this problem, especially when the manifold is a Riemannian product. Also we are going to show multiplicity results when M = S n × S k is a product of spheres. Isoparametric hypersurfaces and the Yamabe equation

Geometric meaning of the Yamabe equation: The conformal class of g is [ g ] := { fg : f ∈ C ∞ > 0 ( M ) } ⊆ M ( M ) . ⇒ h = u p n − 2 g with u a h ∈ [ g ] has constant scalar curvature c ⇐ h has constant scalar curvature c ⇐ ⇒ positive solution of h has constant scalar curvature c ⇐ ⇒ the Yamabe equation. a n ∆ g u + s g u = cu p n − 1 Isoparametric hypersurfaces and the Yamabe equation

Constant scalar curvature metrics in [ g ] ⇐ ⇒ positive solutions of Constant scalar curvature metrics in [ g ] ⇐ ⇒ the Yamabe equation. A function f ∈ C ∞ ( M ) is the scalar curvature of some Riemannian metric iff there exists g ∈ M ( M ) and u ∈ C ∞ > 0 ( M ) such that a n ∆ g u + s g u = f u p n − 1 . In that case, the scalar curvature of the metric u p n − 2 g is s u pn − 2 g = f Isoparametric hypersurfaces and the Yamabe equation

Variational point of view Scalar curvature functional: � M s h dv h h ∈ M ( M ) − → J ( h ) = . n − 2 Vol ( M , h ) n The critical points are the Einstein metrics ( J is not bounded). If we restrict J to a conformal class [ g ], the critical points are the constant scalar curvature metrics in [ g ]. Y = J | [ g ] is called the (Yamabe functional) and is bounded from below. The conformal class [ g ] is parametrized by C ∞ > 0 ( M ): h ∈ [ g ] ∼ u ∈ C ∞ → h = u p n − 2 g . > 0 ( M ) ← M a n �∇ u � 2 + s g u 2 dv g � u − → Y ( u ) = . � u � 2 p n Isoparametric hypersurfaces and the Yamabe equation

The Yamabe equation a n ∆ g u + s g u = cu p n − 1 is the Euler-Lagrange equation of � M a n �∇ u � 2 + s g u 2 dv g u − → Y ( u ) = . � u � 2 p n Positive solutions of the Yamabe equation ⇐ ⇒ positive critical points of Y ⇐ ⇒ metrics of constant scalar curvature in [ g ]. Isoparametric hypersurfaces and the Yamabe equation

The Yamabe constant is defined by Y ( M , [ g ]) := inf h ∈ [ g ] Y ( h ) = 1 ( M ) −{ 0 } Y ( f ) inf f ∈ H 2 2 Y ( M , [ g ]) ≤ Y ( S n , [ g n 0 ]) = Y ( g n 0 ) = n ( n − 1) vol ( S n ) n . Actually, Y ( M , [ g ]) is a minimum = ⇒ Y has at least one Actually, Y ( M , [ g ]) is a minimum = ⇒ critical point in [ g ]. Yamabe Problem In any conformal class there is at least a metric of constant scalar curvature (of a given volume). H. Yamabe (1960), N. Tr¨ udinger (1968), T. Aubin (1976), R. Schoen (1984) Isoparametric hypersurfaces and the Yamabe equation

Meaning of the Yamabe constant The Yamabe constant determines the sign of the scalar curvature in the conformal class. • Y ( M , [ g ]) > 0 iff there exist h ∈ [ g ] with s h > 0. • Y ( M , [ g ]) = 0 iff there exist h ∈ [ g ] with s h ≡ 0. • Y ( M , [ g ]) < 0 iff there exist h ∈ [ g ] with s h < 0. Isoparametric hypersurfaces and the Yamabe equation

Meaning of the Yamabe constant The Yamabe constant determines the sign of the scalar curvature in the conformal class. • Y ( M , [ g ]) > 0 iff there exist h ∈ [ g ] with s h > 0. • Y ( M , [ g ]) = 0 iff there exist h ∈ [ g ] with s h ≡ 0. • Y ( M , [ g ]) < 0 iff there exist h ∈ [ g ] with s h < 0. The Yamabe equation admits a positive solution u a n ∆ g u + s g u = c | u | p n − 2 u if and only if � � sign ( c ) = sign Y ( M , [ g ]) . Isoparametric hypersurfaces and the Yamabe equation

Multiplicity of solutions of the Yamabe equation. If Y ( M , [ g ]) ≤ 0 there is essentially only one positive solution of the Yamabe equation. If ( M , g ) is not conformal to ( S n , g n 0 ) and there is an Einstein metric in [ g ], then there is essentially only one positive solution of the Yamabe equation (the Einstein metric) [Obata] If Y ( M , [ g ]) > 0, could exist several positive solutions (of a given volume). Isoparametric hypersurfaces and the Yamabe equation

An example Let ( M m , g ) and ( N n , h ) be two closed Riemannian manifolds of constant positive scalar curvature and unit volume. ( M × N , t m g + t − n h ) has constant scalar curvature t − m s g + t n s h and unit volume. Therefore, t →∞ or t → + ∞ Y ( t m g + t − n h ) = t − m s g + t n s h − → + ∞ Fot t big enough (or t small enough), Y ( t m g + t − n h ) > Y ( S m + n , [ g m + n ]) ≥ Y ( M × N , [ t m g + t − n h ]) . 0 Then the constant scalar curvature t m g + t − n h does not miminize Y in [ t m g + t − n h ]. There exists another constant scalar curvature in [ t m g + t − n h ]! Isoparametric hypersurfaces and the Yamabe equation

Positive solutions for ( S n , g n 0 ). The space of positive solutions of the Yamabe equation on the standard sphere is non-compact. Metrics of constant scalar curvature in [ g n 0 ] are of the form � n − 2 � ε c ψ p n − 2 g n 2 0 , with ψ ε ( x ) = ε 2 + d 2 ( x , x 0 ) ǫ Isoparametric hypersurfaces and the Yamabe equation

Khuri, Marques and Schoen (2009) If ( M n , g ) is not conformal to ( S n , g n 0 ) and n = dim M ≤ 24, then space of positive solutions of the Yamabe equation is compact in the C 2 − topology. Brendle (2008), Brendle and Marques (2009) In each dimension n ≥ 25 there exist examples where the space of solutions is not compact. Isoparametric hypersurfaces and the Yamabe equation

Why study solutions of Yamabe equation in products? The Yamabe invariant is an invariant of the differential structure defined by Y ( M ) := sup Y ( M , [ g ]) [ g ] ∈C Y ( M ) > 0 ⇐ ⇒ there exists g ∈ M ( M ) with s g > 0 . Isoparametric hypersurfaces and the Yamabe equation

Why study solutions of Yamabe equation in products? The Yamabe invariant is an invariant of the differential structure defined by Y ( M ) := sup Y ( M , [ g ]) [ g ] ∈C Y ( M ) > 0 ⇐ ⇒ there exists g ∈ M ( M ) with s g > 0 . 0 ]) and few more Y ( S n × S 1 ) = Y ( S n +1 ) Y ( S n ) = Y ( S n , [ g n 2 H 3 / Γ) = − 6 vol ( I H 3 / Γ , g 3 3 (Anderson), Y ( T n ) = 0 (Schoen), Y ( I h ) P 2 ) = 12 √ π (LeBrun), (Gromov et al.), Y ( C I R P 3 ) = 2 − 2 3 Y ( S 3 ) (Bray-Neves). Y ( I But not for Y ( S 2 × S 2 ) =?. Isoparametric hypersurfaces and the Yamabe equation

Kazdan and Warner (1975): • Y ( M ) > 0 iff for any f ∈ C ∞ ( M ) there exists h ∈ M ( M ) such that s h = f . • Y ( M ) = 0 and there exists g 0 such that Y ( M , [ g 0 ]) = 0, Then f = s g iif f ≡ 0 o f ( p 0 ) < 0 for some p 0 ∈ M . • Y ( M ) < 0 or Y ( M ) = 0 but the invariant is not realized, f = s g iif f ( p 0 ) < 0 for some p 0 ∈ M . Isoparametric hypersurfaces and the Yamabe equation

Amman, Dahl and Humbert (2013): If N is obtained from M n by performing surgeries of co-dimension n − k ≥ 3, then Y ( N ) ≥ min { Y ( M ) , Λ n , k } If n − k ≥ 4 or n − k = 3 and n = 4 , 5 , 6: H k +1 × S n − k − 1 , [ g k +1 + g n − k − 1 Λ n , k = c ∈ [0 , 1] Y ( I inf ]) 0 h , c where g k +1 is the hyperbolic metric of curvature − c 2 . h , c Y ( S n × I T → 0 Y ( S n × S k , [ Tg n R k , [ g n 0 + g k 0 + g k e ]) = lim 0 ]) [Akutagawa, Florit and Petean (2007)] Isoparametric hypersurfaces and the Yamabe equation

Amman, Dahl and Humbert (2013): If N is obtained from M n by performing surgeries of co-dimension n − k ≥ 3, then Y ( N ) ≥ min { Y ( M ) , Λ n , k } If n − k ≥ 4 or n − k = 3 and n = 4 , 5 , 6: H k +1 × S n − k − 1 , [ g k +1 + g n − k − 1 Λ n , k = c ∈ [0 , 1] Y ( I inf ]) 0 h , c where g k +1 is the hyperbolic metric of curvature − c 2 . h , c Y ( S n − k − 1 × I T → 0 Y ( S n − k − 1 × S k +1 , [ Tg 0 + g 0 ]) R k +1 , [ g 0 + g e ]) = lim [Akutagawa, Florit and Petean (2009)] Isoparametric hypersurfaces and the Yamabe equation