The Asia-Pacifjc Analysis and PDE Seminar New Sharp Inequalities in - PowerPoint PPT Presentation

D n e u S 1 ud u 2 D n e u S 1 ud L 2 D u 2 D n e u S 1 ud L 2 D 1 1 n Then Defjne Theorem New Sharp Toeplitz Determinants and the Szego Limit . . . . . . Inequalities in . 2 n is nondecreasing and 1 0 n 4 1 2 1 n n In particular, 4 1 2 1 1 . . . . Inequality . . . . . . . . New Inequality Determinants Logrithemic Covering . Sphere Inequality Aubin-Onofri Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . . . . . . . . . . . . . . . . . . . . . Given f ( θ ) ∈ L 1 ( S 1 ) . Let ∫ S 1 e ik θ f ( θ ) d θ, k = 0 , ± 1 , ± 2 , · · · , c k = 1 2 π and T ( p , q ) = c p − q , p , q ∈ Z be the Toeplitz matrix, and T n ( p , q ) = c p − q , 0 ≤ p , q ≤ n be the n-th Toeplitz matrix. D n ( f ) = det ( T n ) .

u 2 D n e u S 1 ud L 2 D New Sharp Inequalities in . . . . . . . . . . . . . . . . . . Toeplitz Determinants and the Szego Limit Theorem Defjne In particular, n 1 1 2 1 4 n 0 . . . Aubin-Onofri . Logrithemic Inequality Covering Sphere Inequality Determinants New Inequality Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and Determinants . . . . . . . . . . . . . . . . . . . Given f ( θ ) ∈ L 1 ( S 1 ) . Let ∫ S 1 e ik θ f ( θ ) d θ, k = 0 , ± 1 , ± 2 , · · · , c k = 1 2 π and T ( p , q ) = c p − q , p , q ∈ Z be the Toeplitz matrix, and T n ( p , q ) = c p − q , 0 ≤ p , q ≤ n be the n-th Toeplitz matrix. D n ( f ) = det ( T n ) . ∫ Then ln D n ( e u ) − ( n + 1 ) 1 S 1 ud θ is nondecreasing and 2 π ∫ lim n →∞ { ln D n ( e u ) − ( n + 1 ) 1 S 1 ud θ } = 1 4 π ||∇ u || 2 L 2 ( D ) . 2 π

New Sharp . . . . . . . . . . . Inequalities in . . . . . . . . . . . . . Toeplitz Determinants and the Szego Limit Theorem Defjne In particular, . . . Inequality Analysis and Geometry . Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering . Logrithemic . New Inequality . . . . . . . . . . . Determinants Given f ( θ ) ∈ L 1 ( S 1 ) . Let ∫ S 1 e ik θ f ( θ ) d θ, k = 0 , ± 1 , ± 2 , · · · , c k = 1 2 π and T ( p , q ) = c p − q , p , q ∈ Z be the Toeplitz matrix, and T n ( p , q ) = c p − q , 0 ≤ p , q ≤ n be the n-th Toeplitz matrix. D n ( f ) = det ( T n ) . ∫ Then ln D n ( e u ) − ( n + 1 ) 1 S 1 ud θ is nondecreasing and 2 π ∫ lim n →∞ { ln D n ( e u ) − ( n + 1 ) 1 S 1 ud θ } = 1 4 π ||∇ u || 2 L 2 ( D ) . 2 π ∫ ln D n ( e u ) − ( n + 1 ) 1 S 1 ud θ ≤ 1 4 π ||∇ u || 2 L 2 ( D ) , n ≥ 0 . 2 π

S 1 e u e i d u 2 S 1 e u d S 1 ud L 2 D S 1 e u e i d u 2 S 1 e u d S 1 ud L 2 D 2 1 1, the second inequality in the Szego limit theorem is When n Lebedev-Milin Inequality. New Sharp We have The fjrst two inequalities in the Szego limit theorem 2 . . . . . Inequalities in . . 1 1 1 (3) 8 1 2 1 2 direct consequence of above inequality we have 2 0, as a One notes that in the special case when (2) 4 1 . 2 . . . Inequality . . . . . . . . New Inequality Determinants Logrithemic Covering . Sphere Inequality Aubin-Onofri Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . . . . . . . . . . . . . . Question: Any similar inequalities in higher dimensions? . . . . . . ∫ ∫ S 1 e u d θ ) 2 − ( 1 S 1 e u e i θ d θ ) 2 . D 1 ( e u ) = ( 1 2 π 2 π The fjrst inequality when n = 0 of Szego limit theorem is the

S 1 e u e i d u 2 S 1 e u d S 1 ud L 2 D Inequalities in . . . . . New Sharp . . . . . . . . . We have . The fjrst two inequalities in the Szego limit theorem . Lebedev-Milin Inequality. (2) One notes that in the special case when 0, as a direct consequence of above inequality we have 1 2 1 2 1 8 (3) . . . Aubin-Onofri New Inequality Determinants Logrithemic Inequality Covering Sphere Inequality Determinants . Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . Question: Any similar inequalities in higher dimensions? . . . . . . . . . . . . . . . . . . . ∫ ∫ S 1 e u d θ ) 2 − ( 1 S 1 e u e i θ d θ ) 2 . D 1 ( e u ) = ( 1 2 π 2 π The fjrst inequality when n = 0 of Szego limit theorem is the When n = 1, the second inequality in the Szego limit theorem is ∫ ∫ ∫ S 1 e u e i θ d θ | 2 ) − 1 log( | 1 S 1 e u d θ | 2 −| 1 S 1 ud θ ≤ 1 4 π ||∇ u || 2 L 2 ( D ) . 2 π 2 π π

New Sharp Inequalities in . . . . . . . . . . . . . . . . . . . . . . The fjrst two inequalities in the Szego limit theorem We have Lebedev-Milin Inequality. (2) One notes that in the special case when direct consequence of above inequality we have (3) . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin . Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Question: Any similar inequalities in higher dimensions? Determinants . . . . . . . . . . . . . . New Inequality ∫ ∫ S 1 e u d θ ) 2 − ( 1 S 1 e u e i θ d θ ) 2 . D 1 ( e u ) = ( 1 2 π 2 π The fjrst inequality when n = 0 of Szego limit theorem is the When n = 1, the second inequality in the Szego limit theorem is ∫ ∫ ∫ S 1 e u e i θ d θ | 2 ) − 1 log( | 1 S 1 e u d θ | 2 −| 1 S 1 ud θ ≤ 1 4 π ||∇ u || 2 L 2 ( D ) . 2 π 2 π π S 1 e u e i θ d θ = 0, as a ∫ ∫ ∫ log( 1 S 1 e u d θ ) − 1 S 1 ud θ ≤ 1 8 π ||∇ u || 2 L 2 ( D ) . 2 π 2 π

New Sharp Inequalities in . . . . . . . . . . . . . . . . . . . . . . The fjrst two inequalities in the Szego limit theorem We have Lebedev-Milin Inequality. (2) One notes that in the special case when direct consequence of above inequality we have (3) . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin . Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Question: Any similar inequalities in higher dimensions? Determinants . . . . . . . . . . . . . . New Inequality ∫ ∫ S 1 e u d θ ) 2 − ( 1 S 1 e u e i θ d θ ) 2 . D 1 ( e u ) = ( 1 2 π 2 π The fjrst inequality when n = 0 of Szego limit theorem is the When n = 1, the second inequality in the Szego limit theorem is ∫ ∫ ∫ S 1 e u e i θ d θ | 2 ) − 1 log( | 1 S 1 e u d θ | 2 −| 1 S 1 ud θ ≤ 1 4 π ||∇ u || 2 L 2 ( D ) . 2 π 2 π π S 1 e u e i θ d θ = 0, as a ∫ ∫ ∫ log( 1 S 1 e u d θ ) − 1 S 1 ud θ ≤ 1 8 π ||∇ u || 2 L 2 ( D ) . 2 π 2 π

New Sharp . . . . . . . . . . . . . . . . . . . . . . . . Outline 1 Lebedev-Milin Inequality and Toeplitz Determinants 2 Aubin-Onofri Inequality 3 Sphere Covering Inequality 4 Logrithemic Determinants . . Inequalities in Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Determinants . New Inequality . . . . . . . . . . . . . 5 New Inequality

New Sharp . . . Inequalities in . . . . . . . . . . . . . . . . . . . . . Trudinger-Moser Inequality (1967, 1971) 4 so that . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri . Sphere Covering Inequality Inequality Determinants . . . . . . . . . . . New Inequality . Let S 2 be the unit sphere and for u ∈ H 1 ( S 2 ) . J α ( u ) = α ∫ ∫ ∫ S 2 |∇ u | 2 d ω + S 2 ud ω − log S 2 e u d ω ≥ C > −∞ , if and only if α ≥ 1, where the volume form d ω is normalized ∫ S 2 d ω = 1.

H 1 S 2 S 2 e u x i New Sharp . . . . . . . . . . . . . . . . . u i 0 u u for C J Inequalities in 2 , 1 Aubin observed that for Aubin’s Result (1979) and Onofri Inequality (1982) . . . . . Determinants Logrithemic Inequality Covering Sphere Inequality Aubin-Onofri Toeplitz . Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and Determinants New Inequality . . . . . . . . . . . . . . . . . 1 2 3 Onofri showed for α ≥ 1 J α ( u ) ≥ 0 ;

New Sharp . Inequalities in . . . . . . . . . . . . . . . . . . . . . . . Aubin’s Result (1979) and Onofri Inequality (1982) 2 , for . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality . Covering Inequality Sphere Determinants . . . New Inequality . . . . . . . . . Onofri showed for α ≥ 1 J α ( u ) ≥ 0 ; Aubin observed that for α ≥ 1 J α ( u ) ≥ C > −∞ ∫ u ∈ M := { u ∈ H 1 ( S 2 ) : S 2 e u x i = 0 , i = 1 , 2 , 3 } ,

S 2 e u d New Sharp . 1 Conjecture A. For They proposed the following conjecture. again is equal to zero. Chang and Yang Conjecture (1987) . . . u . . . . . . . . 2 , u J i i 0 i and (4) on S 2 i x i e u 1 3 Inequalities in i 1 e u u 2 the Euler-Lagrange equations Indeed, they showed that the minimizer u exists and satisfjes 0 . . . Inequality . . New Inequality Determinants Logrithemic Inequality Covering Sphere Aubin-Onofri . Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . . . . . . . . . . . . . . . . . . . . . . 1 2 3 Chang and Yang showed that for α close to 1 the best constant

S 2 e u d New Sharp . Chang and Yang Conjecture (1987) . . . . . . They proposed the following conjecture. . . . . . . . . again is equal to zero. 2 , . 1 i 0 i and (4) on S 2 i x i e u i Indeed, they showed that the minimizer u exists and satisfjes 3 i 1 e u u 2 the Euler-Lagrange equations Inequalities in . . Inequality . New Inequality Determinants Logrithemic Inequality Covering Sphere Aubin-Onofri . Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . . 1 2 3 . . . . . . . . . . . . . . . . . . Chang and Yang showed that for α close to 1 the best constant Conjecture A. For α ≥ 1 inf u ∈M J α ( u ) = 0 .

New Sharp . . . . . . . . . . Inequalities in . . . . . . . the Euler-Lagrange equations i 0 i and (4) e u Indeed, they showed that the minimizer u exists and satisfjes . 2 , They proposed the following conjecture. again is equal to zero. Chang and Yang Conjecture (1987) . . . . . Determinants Logrithemic Inequality Covering . Inequality Aubin-Onofri Toeplitz New Inequality Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and Determinants Sphere . . . . . . . . . . 1 2 3 . . . . . . Chang and Yang showed that for α close to 1 the best constant Conjecture A. For α ≥ 1 inf u ∈M J α ( u ) = 0 . i = 3 α ∑ S 2 e u d ω − 1 = 2 ∆ u + µ i x i e u , on S 2 . ∫ i = 1

New Sharp . . . . . Inequalities in . . . . . . . . . . . . . . . . Chang and Yang Conjecture (1987) again is equal to zero. They proposed the following conjecture. 2 , Indeed, they showed that the minimizer u exists and satisfjes the Euler-Lagrange equations e u (4) and . . . Determinants Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants . Inequality Sphere Covering Inequality Logrithemic Aubin-Onofri New Inequality . . . . . . . . . . . . . . . Chang and Yang showed that for α close to 1 the best constant Conjecture A. For α ≥ 1 inf u ∈M J α ( u ) = 0 . i = 3 α ∑ S 2 e u d ω − 1 = 2 ∆ u + µ i x i e u , on S 2 . ∫ i = 1 µ i = 0 , i = 1 , 2 , 3 .

New Sharp . . . Inequalities in . . . . . . . . . . . . . . . . . . . . Chang and Yang Conjecture (1987) again is equal to zero. They proposed the following conjecture. 2 , Indeed, they showed that the minimizer u exists and satisfjes e u . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality . Covering Inequality Sphere Determinants . . . . New Inequality . . . . . . . (5) . . Chang and Yang showed that for α close to 1 the best constant Conjecture A. For α ≥ 1 inf u ∈M J α ( u ) = 0 . α S 2 e u d ω − 1 = 0 , 2 ∆ u + on S 2 . ∫

x 2 g x e 2 g x dx New Sharp . . . . . . . . it holds for . . . . . . Inequalities in Axially symmetric functions 2 1 2, 2 25 16 Feldman, Froese, Ghoussoub and G. (1998) 0 1 1 1 . g x dx 1 1 2 dx 1 1 1 . . . Sphere . . . New Inequality Determinants Logrithemic Inequality Covering Inequality . Aubin-Onofri Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . G. and Wei, and independently Lin (2000) . . . . . . . . . . . . . . . . . For every function g on ( − 1 , 1 ) satisfying ∥ g ∥ 2 = − 1 ( 1 − x 2 ) | g ′ ( x ) | 2 dx < ∞ and ∫ 1 ∫ 1 e 2 g ( x ) xdx = 0 , − 1

New Sharp . . . . . . . Inequalities in . . . . . . . . . . . . . . . . Axially symmetric functions 2 2 Feldman, Froese, Ghoussoub and G. (1998) 16 25 . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin . Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Inequality and Determinants . . . . . . New Inequality G. and Wei, and independently Lin (2000) . . . . . . . For every function g on ( − 1 , 1 ) satisfying ∥ g ∥ 2 = − 1 ( 1 − x 2 ) | g ′ ( x ) | 2 dx < ∞ and ∫ 1 ∫ 1 e 2 g ( x ) xdx = 0 , − 1 it holds for α ≥ 1 / 2, α ∫ 1 ∫ 1 ∫ 1 ( 1 − x 2 ) | g ′ ( x ) | 2 dx + e 2 g ( x ) dx ≥ 0 , g ( x ) dx − log 1 − 1 − 1 − 1

New Sharp . . . . . . . . Inequalities in . . . . . . . . . . . . . . . . Axially symmetric functions 2 2 Feldman, Froese, Ghoussoub and G. (1998) . . . Logrithemic Analysis and Geometry Changfeng Gui . Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Lebedev-Milin Determinants . . . . New Inequality . G. and Wei, and independently Lin (2000) . . . . . . . For every function g on ( − 1 , 1 ) satisfying ∥ g ∥ 2 = − 1 ( 1 − x 2 ) | g ′ ( x ) | 2 dx < ∞ and ∫ 1 ∫ 1 e 2 g ( x ) xdx = 0 , − 1 it holds for α ≥ 1 / 2, α ∫ 1 ∫ 1 ∫ 1 ( 1 − x 2 ) | g ′ ( x ) | 2 dx + e 2 g ( x ) dx ≥ 0 , g ( x ) dx − log 1 − 1 − 1 − 1 α > 16 25 − ϵ

New Sharp . . . . . . . . Inequalities in . . . . . . . . . . . . . . . . Axially symmetric functions 2 2 Feldman, Froese, Ghoussoub and G. (1998) . . . Logrithemic Analysis and Geometry Changfeng Gui . Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Lebedev-Milin Determinants . . . . New Inequality . G. and Wei, and independently Lin (2000) . . . . . . . For every function g on ( − 1 , 1 ) satisfying ∥ g ∥ 2 = − 1 ( 1 − x 2 ) | g ′ ( x ) | 2 dx < ∞ and ∫ 1 ∫ 1 e 2 g ( x ) xdx = 0 , − 1 it holds for α ≥ 1 / 2, α ∫ 1 ∫ 1 ∫ 1 ( 1 − x 2 ) | g ′ ( x ) | 2 dx + e 2 g ( x ) dx ≥ 0 , g ( x ) dx − log 1 − 1 − 1 − 1 α > 16 25 − ϵ

New Sharp . . . . . . . . . . . . . . . . . . . . . . . . . Earlier Result for general functions: Ghoussoub and Lin (2010): Conjecture A holds for . . Inequalities in . Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Logrithemic Determinants New Inequality . . . . . . . . . . . . α ≥ 2 3 − ϵ

New Sharp . . . . . . . . . . . . . . . . . . . . . . . . . Strategies of Proof For axially symmetric functions, to show (3) has only solution For general functions, to show solutions to (3) are axially . . Inequalities in . Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Logrithemic Determinants New Inequality . . . . . . . . . . . . symmetric. u ≡ C .

New Sharp . . . . . . . . . . . . . . . . . . . . . . . . . Strategies of Proof For axially symmetric functions, to show (3) has only solution For general functions, to show solutions to (3) are axially . . Inequalities in . Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Logrithemic Determinants New Inequality . . . . . . . . . . . . symmetric. u ≡ C .

New Sharp . . . . . . . . . . . . . . . . . . . . . . . . . . Sterographic Projection . . Inequalities in Inequality Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Logrithemic . Determinants New Inequality . . . . . . . . . . . Figure: Sterographic Projection

1 y y 2 y 2 2 1 e v y 2 2 1 e v dy R 2 1 New Sharp . x 1 . . . . . . Suppose u is a solution of (3) and let . . . Inequalities in . . x 2 2 v 1 8 1 and (7) 2 0 in 1 u v then v satisfjes (6) 8 1 . . . . Covering . . . . . . New Inequality Determinants Logrithemic Inequality Sphere . Inequality Aubin-Onofri Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . (8) . . . . . . . . . . . . . . . . Equations on R 2 Let Π be the stereographic projection S 2 → R 2 with respect to the north pole N = ( 1 , 0 , 0 ) : ( ) Π := , . 1 − x 3 1 − x 3

y 2 2 1 e v y 2 2 1 e v dy R 2 1 New Sharp . . . . . . . . . . Inequalities in . . . . x 2 . x 1 . Suppose u is a solution of (3) and let (6) then v satisfjes v 1 1 0 in 2 (7) and 1 8 . . . Inequality . . New Inequality Determinants Logrithemic Inequality Covering Sphere Aubin-Onofri . Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . (8) . . . . . . . . . . . . . . . . . Equations on R 2 Let Π be the stereographic projection S 2 → R 2 with respect to the north pole N = ( 1 , 0 , 0 ) : ( ) Π := , . 1 − x 3 1 − x 3 v = u (Π − 1 ( y )) − 2 α ln( 1 + | y | 2 ) + ln( 8 α ) ,

New Sharp . . . . . . . Inequalities in . . . . . . . . . . . . . . . x 1 x 2 Suppose u is a solution of (3) and let (6) then v satisfjes (7) and . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz . Determinants Aubin-Onofri Inequality Sphere Covering Inequality (8) Determinants . . . . . . . . New Inequality . . . . . . Equations on R 2 Let Π be the stereographic projection S 2 → R 2 with respect to the north pole N = ( 1 , 0 , 0 ) : ( ) Π := , . 1 − x 3 1 − x 3 v = u (Π − 1 ( y )) − 2 α ln( 1 + | y | 2 ) + ln( 8 α ) , α − 1 ) e v = 0 in R 2 , ∆ v + ( 1 + | y | 2 ) 2 ( 1 α − 1 ) e v dy = 8 π ∫ R 2 ( 1 + | y | 2 ) 2 ( 1 α .

New Sharp . . . . . . . . . . . . . . . . . . 0: Chen and Li (1991) l 0 0: Chanillo and Kiessling (1994) l 2 For For l . Are solutions to (9) and (10) radially symmetric? (10) and (9) Consider in general the equation General Equations . Inequalities in . . Determinants Logrithemic Inequality Covering Sphere Inequality Aubin-Onofri Toeplitz . Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and Determinants New Inequality . . . . . . . . . . . . . . . . . 1: Ghoussoub and Lin (2010) ∆ v + ( 1 + | y | 2 ) l e v = 0 in R 2 , ∫ R 2 ( 1 + | y | 2 ) l e v dy = 2 π ( 2 l + 4 ) .

New Sharp . . . . . . . . . . . . . . . . . . 0: Chen and Li (1991) l 0 0: Chanillo and Kiessling (1994) l 2 For For l . Are solutions to (9) and (10) radially symmetric? (10) and (9) Consider in general the equation General Equations . Inequalities in . . Determinants Logrithemic Inequality Covering Sphere Inequality Aubin-Onofri Toeplitz . Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and Determinants New Inequality . . . . . . . . . . . . . . . . . 1: Ghoussoub and Lin (2010) ∆ v + ( 1 + | y | 2 ) l e v = 0 in R 2 , ∫ R 2 ( 1 + | y | 2 ) l e v dy = 2 π ( 2 l + 4 ) .

New Sharp . . . . . . . . . . . . . . . . . . Are solutions to (9) and (10) radially symmetric? l 0 0: Chanillo and Kiessling (1994) l 2 For (10) . and (9) Consider in general the equation General Equations . . Inequalities in . . Determinants Logrithemic Inequality Covering Sphere Inequality Aubin-Onofri Toeplitz New Inequality Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . Determinants . . . . . . . . . 1: Ghoussoub and Lin (2010) . . . . . . . ∆ v + ( 1 + | y | 2 ) l e v = 0 in R 2 , ∫ R 2 ( 1 + | y | 2 ) l e v dy = 2 π ( 2 l + 4 ) . For l = 0: Chen and Li (1991)

New Sharp . Inequalities in . . . . . . . . . . . . . . . . . . . . . General Equations Consider in general the equation (9) and (10) Are solutions to (9) and (10) radially symmetric? 0 l . . . Determinants Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering . Logrithemic Inequality New Inequality . . . . . . . . . . . . . . 1: Ghoussoub and Lin (2010) ∆ v + ( 1 + | y | 2 ) l e v = 0 in R 2 , ∫ R 2 ( 1 + | y | 2 ) l e v dy = 2 π ( 2 l + 4 ) . For l = 0: Chen and Li (1991) For − 2 < l < 0: Chanillo and Kiessling (1994)

New Sharp . Inequalities in . . . . . . . . . . . . . . . . . . . . . . General Equations Consider in general the equation (9) and (10) Are solutions to (9) and (10) radially symmetric? . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering . Inequality Determinants . . . . . New Inequality . . . . . . . . ∆ v + ( 1 + | y | 2 ) l e v = 0 in R 2 , ∫ R 2 ( 1 + | y | 2 ) l e v dy = 2 π ( 2 l + 4 ) . For l = 0: Chen and Li (1991) For − 2 < l < 0: Chanillo and Kiessling (1994) 0 < l ≤ 1: Ghoussoub and Lin (2010)

New Sharp . . . . . . . . . . . . . . . . . . Inequalities in 2 distinct radial 2, solutions to (9) and (10) must l Conjecture B. For 0 structure becomes.) (The bigger the l is, the more complicated the solution solutions, which implies the existence of non radial solutions. 2 Existence of Non Radial Solutions 2, there are at least 2 k 1 k k l 2 and Dolbeault, Esteban, Tarantello (2009): For all k non radial solution. . . . Aubin-Onofri Determinants Logrithemic Inequality Covering Sphere Inequality Determinants . Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and New Inequality . . . . . . . . . . . . . . . . . . be radially symmetric. Lin (2000): For 2 < l ̸ = ( k − 1 )( k + 2 ) , where k ≥ 2 there is a

New Sharp . . . . . . . . . . . . . . . . . . . . . . . Existence of Non Radial Solutions non radial solution. solutions, which implies the existence of non radial solutions. (The bigger the l is, the more complicated the solution structure becomes.) Conjecture B. For 0 l 2, solutions to (9) and (10) must Inequalities in . . Determinants Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality . Logrithemic New Inequality . . . . . . . . . . . . . . be radially symmetric. Lin (2000): For 2 < l ̸ = ( k − 1 )( k + 2 ) , where k ≥ 2 there is a Dolbeault, Esteban, Tarantello (2009): For all k ≥ 2 and l > k ( k + 1 ) − 2, there are at least 2 ( k − 2 ) + 2 distinct radial

New Sharp . . . . . . . . . . . . . . . . . . . . . . . . Existence of Non Radial Solutions non radial solution. solutions, which implies the existence of non radial solutions. (The bigger the l is, the more complicated the solution structure becomes.) Inequalities in . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering . Inequality Determinants . . . . . New Inequality . . . . . . . . be radially symmetric. Lin (2000): For 2 < l ̸ = ( k − 1 )( k + 2 ) , where k ≥ 2 there is a Dolbeault, Esteban, Tarantello (2009): For all k ≥ 2 and l > k ( k + 1 ) − 2, there are at least 2 ( k − 2 ) + 2 distinct radial Conjecture B. For 0 < l ≤ 2, solutions to (9) and (10) must

New Sharp . . . . . . . . . . . . . . . . . . . . . . . . Existence of Non Radial Solutions non radial solution. solutions, which implies the existence of non radial solutions. (The bigger the l is, the more complicated the solution structure becomes.) Inequalities in . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering . Inequality Determinants . . . . . New Inequality . . . . . . . . be radially symmetric. Lin (2000): For 2 < l ̸ = ( k − 1 )( k + 2 ) , where k ≥ 2 there is a Dolbeault, Esteban, Tarantello (2009): For all k ≥ 2 and l > k ( k + 1 ) − 2, there are at least 2 ( k − 2 ) + 2 distinct radial Conjecture B. For 0 < l ≤ 2, solutions to (9) and (10) must

2 1 2 8 New Sharp . . . . . . . . . 2018) . . . . . . Main Theorem ( G. and Moradifam, Inventiones, Conjecture A. Both Conejcture A and B hold true. For 0 1 1 l Note symmetric. 2, solutions to (9) and (10) must be radially l Conjecture B. . 0 u J u 2 , 1 For . . Inequalities in Aubin-Onofri New Inequality Determinants Logrithemic Inequality Covering Sphere Inequality Determinants . Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . . . . . . . . . . . . . . . . . . . . . .

2 1 2 8 New Sharp . . . . . . . . . . . . . . . . . . l 1 1 l Note symmetric. 2, solutions to (9) and (10) must be radially For 0 . Conjecture B. 2 , Conjecture A. Both Conejcture A and B hold true. 2018) Main Theorem ( G. and Moradifam, Inventiones, Inequalities in . . Aubin-Onofri Determinants Logrithemic Inequality Covering Sphere Inequality Determinants . Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . New Inequality . . . . . . . . . . . . . . . . . . For α ≥ 1 inf u ∈M J α ( u ) = 0 .

2 1 2 8 New Sharp . . . . . . . . . . . . . . . . . . . . Main Theorem ( G. and Moradifam, Inventiones, 2018) Both Conejcture A and B hold true. Conjecture A. 2 , Conjecture B. symmetric. Note l 1 1 Inequalities in . . Determinants Logrithemic Inequality Covering Sphere Inequality Aubin-Onofri Toeplitz New Inequality Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . Determinants . . . . . . . . . . . . . . . . . . For α ≥ 1 inf u ∈M J α ( u ) = 0 . For 0 < l ≤ 2, solutions to (9) and (10) must be radially

New Sharp . . Inequalities in . . . . . . . . . . . . . . . . . . . . Main Theorem ( G. and Moradifam, Inventiones, 2018) Both Conejcture A and B hold true. Conjecture A. 2 , Conjecture B. symmetric. Note . . . Determinants Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere . Inequality Logrithemic Covering New Inequality . . . . . . . . . . . . . . . For α ≥ 1 inf u ∈M J α ( u ) = 0 . For 0 < l ≤ 2, solutions to (9) and (10) must be radially α − 1 ) = 2 ( ρ l = 2 ( 1 8 π − 1 )

2 k y e 2 u dy 2 is a non constant positive New Sharp 1 and (11) A general equation on R 2 . . . . . . . . . . . . . 2 (12) Inequalities in K 2 y 0 2 l k y y y k y 2 where K y y 0 k y K 1 function satisfying C 2 k y . . . Inequality . . New Inequality Determinants Logrithemic Inequality Covering Sphere Aubin-Onofri . Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . . . . . . . . . . . . . . . . . . . . . 2 Assume u ∈ C 2 ( R 2 ) satisfjes ∆ u + k ( | y | ) e 2 u = 0 in R 2 ,

New Sharp . . . . . . . . A general equation on R 2 . . . . . . Inequalities in . (11) . K 2 y 0 2 l k y y y k y 2 and y 0 k y K 1 function satisfying (12) 1 . . . Aubin-Onofri New Inequality Determinants Logrithemic Inequality Covering Sphere Inequality Determinants . Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . . 2 . . . . . . . . . . . . . . . . . . Assume u ∈ C 2 ( R 2 ) satisfjes ∆ u + k ( | y | ) e 2 u = 0 in R 2 , ∫ R 2 k ( | y | ) e 2 u dy = β < ∞ , 2 π where K ( y ) = k ( | y | ) ∈ C 2 ( R 2 ) is a non constant positive

New Sharp . . . . . . Inequalities in . . . . . . . . . . . . . . . . . A general equation on R 2 (11) and 1 (12) function satisfying . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz . Aubin-Onofri Inequality Sphere Covering Inequality Determinants Determinants . New Inequality . . . . . . . . . . . . Assume u ∈ C 2 ( R 2 ) satisfjes ∆ u + k ( | y | ) e 2 u = 0 in R 2 , ∫ R 2 k ( | y | ) e 2 u dy = β < ∞ , 2 π where K ( y ) = k ( | y | ) ∈ C 2 ( R 2 ) is a non constant positive ∆ ln( k ( | y | )) ≥ 0 , y ∈ R 2 ( K 1 ) | y | k ′ ( | y | ) ( K 2 ) lim = 2 l > 0 , y ∈ R 2 . k ( | y | ) | y |→∞

New Sharp . . . . . . . . . . . . . . Inequalities in . . . . . . . . . . A general symmetry result The following general symmetry result is proven. Proposition . . . . Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Logrithemic Determinants New Inequality . . . . . . . . . . . . radially symmetric. Assume that K ( y ) = k ( | y | ) > 0 satisfjes ( K 1 ) − ( K 2 ) , and u is a solution to (11) - (12) with l + 1 < β ≤ 4 . Then u must be

New Sharp . . . . . . . . . . . . . . . . . . . . . . . . Outline 1 Lebedev-Milin Inequality and Toeplitz Determinants 2 Aubin-Onofri Inequality 3 Sphere Covering Inequality 4 Logrithemic Determinants . . Inequalities in Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Determinants . New Inequality . . . . . . . . . . . . . 5 New Inequality

New Sharp . . . . . . . . . . . . . . . . . . . . . . . . . . The Sphere Covering Inequality: Geometric . . Inequalities in Inequality Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Logrithemic . Determinants New Inequality . . . . . . . . . . . Description

w 1 in w 1 on f 1 in New Sharp . . . . . . . . . . . . . . . . Statement The Sphere Covering Inequality: Analytic 8 e w 2 dy e w 1 , then 0 or f 2 Furthermore if f 1 (14) e w 2 dy . e w 1 , then and w 2 Suppose w 2 (13) Theorem ( G. and Moradifam, Inventiones, 2018) Inequalities in . . Aubin-Onofri New Inequality Determinants Logrithemic Inequality Covering Sphere Inequality Determinants . Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . . . . . . . . . . . . . . . . . . . . 8 . Let Ω be a simply connected subset of R 2 and assume w i ∈ C 2 (Ω) , i = 1 , 2 satisfy ∆ w i + e w i = f i ( y ) , where f 2 ≥ f 1 ≥ 0 in Ω .

f 1 in New Sharp . . . . . . . . . . . . . . Inequalities in . . . 8 e w 2 dy e w 1 , then 0 or f 2 Furthermore if f 1 (14) e w 2 dy . e w 1 (13) Theorem ( G. and Moradifam, Inventiones, 2018) Statement The Sphere Covering Inequality: Analytic . . . . Determinants Logrithemic Inequality Covering Sphere Inequality Aubin-Onofri Toeplitz New Inequality Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . Determinants . . . . . . . . . . . 8 . . . . . . . Let Ω be a simply connected subset of R 2 and assume w i ∈ C 2 (Ω) , i = 1 , 2 satisfy ∆ w i + e w i = f i ( y ) , where f 2 ≥ f 1 ≥ 0 in Ω . Suppose w 2 > w 1 in Ω and w 2 = w 1 on ∂ Ω , then

f 1 in New Sharp . . . . . . . . . . . . Inequalities in . . . . . . . . The Sphere Covering Inequality: Analytic Statement Theorem ( G. and Moradifam, Inventiones, 2018) (13) (14) Furthermore if f 1 0 or f 2 , then e w 1 e w 2 dy . . . Determinants Logrithemic . Covering Sphere Inequality Aubin-Onofri Toeplitz New Inequality Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and Determinants Inequality . . . . . . . . . . 8 . . . . . . . Let Ω be a simply connected subset of R 2 and assume w i ∈ C 2 (Ω) , i = 1 , 2 satisfy ∆ w i + e w i = f i ( y ) , where f 2 ≥ f 1 ≥ 0 in Ω . Suppose w 2 > w 1 in Ω and w 2 = w 1 on ∂ Ω , then ∫ e w 1 + e w 2 dy ≥ 8 π. Ω

New Sharp . . . Inequalities in . . . . . . . . . . . . . . . . . . . . The Sphere Covering Inequality: Analytic Statement Theorem ( G. and Moradifam, Inventiones, 2018) (13) (14) . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri . Sphere Covering Inequality Inequality Determinants . . . . . . . . . . . . New Inequality . Let Ω be a simply connected subset of R 2 and assume w i ∈ C 2 (Ω) , i = 1 , 2 satisfy ∆ w i + e w i = f i ( y ) , where f 2 ≥ f 1 ≥ 0 in Ω . Suppose w 2 > w 1 in Ω and w 2 = w 1 on ∂ Ω , then ∫ e w 1 + e w 2 dy ≥ 8 π. Ω Ω e w 1 + e w 2 dy > 8 π . ∫ Furthermore if f 1 ̸≡ 0 or f 2 ̸≡ f 1 in Ω , then

New Sharp . . . . . . . . . . . . . . . . . . . . . . . . . Rigidity of Two Objects: Seesaw Efgect e w 1 dy vs Inequalities in . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere . Inequality Covering Determinants . . . . New Inequality . . . . . . . . (15) ∫ ∫ e w 2 dy . Ω Ω

New Sharp . . . . . . . . . . . . . . Inequalities in . . . . . . . . . . . Isoperimetric Inequalities . . . . Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Logrithemic Determinants . . . . . . . . . . . New Inequality Suppose Ω ⊂ R 2 , then L 2 ( ∂ Ω) ≥ 4 π A (Ω) Equality holds if and only if Ω is a disk.

New Sharp . . . . . . . . . . . . . . . . . . A A 4 A L 2 i.e., R 2 4 R 2 . A L 2 If the sphere has radius R , then On the standard unit sphere with the metric induced from the Levy’s Isoperimetric inequalities on spheres (1919) . Inequalities in . . Determinants Logrithemic Inequality Covering Sphere Inequality Aubin-Onofri Toeplitz New Inequality Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . Determinants . . . . . . . . . . . . . . . . . R 2 fmat metric of R 3 , L 2 ( ∂ Ω) ≥ A (Ω) ( 4 π − A (Ω) )

New Sharp . . . . Inequalities in . . . . . . . . . . . . . . . . . . . . Levy’s Isoperimetric inequalities on spheres (1919) On the standard unit sphere with the metric induced from the If the sphere has radius R , then i.e., . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants . Inequality Sphere Covering Inequality Aubin-Onofri Determinants . . . . . . . . . . . New Inequality . fmat metric of R 3 , L 2 ( ∂ Ω) ≥ A (Ω) ( 4 π − A (Ω) ) 4 π R 2 − A (Ω) ( ) L 2 ( ∂ Ω) ≥ A (Ω) / R 2 ( 4 π − A (Ω) / R 2 ) L 2 ( ∂ Ω) ≥ A (Ω)

e v ds 2 New Sharp . . . . . . . . . . . . . . . . . 0 4 e 2 v Then 1 with the gaussian curvature K 2 K x e 2 v . v Assume v satisfjes projection, and equip it with a metric conformal to the fmat Alexandrov-Bol’s inequality (1941) . . Inequalities in . . Determinants Logrithemic Inequality Covering Sphere Inequality Aubin-Onofri Toeplitz New Inequality Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . Determinants . . . . . . . . . . . . . . . . . e 2 v In general, we can identify a sphere with R 2 by a stereographic metric of R 2 , i.e., ds 2 = e 2 v ( dx 2 1 + dx 2 2 ) .

e v ds 2 New Sharp . Inequalities in . . . . . . . . . . . . . . . . . . . . . Alexandrov-Bol’s inequality (1941) projection, and equip it with a metric conformal to the fmat Assume v satisfjes Then e 2 v 4 . . . Determinants Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere . Inequality Logrithemic Covering New Inequality . . . . . . . e 2 v . . . . . . . In general, we can identify a sphere with R 2 by a stereographic metric of R 2 , i.e., ds 2 = e 2 v ( dx 2 1 + dx 2 2 ) . ∆ v + K ( x ) e 2 v = 0 , R 2 with the gaussian curvature K ≤ 1 .

New Sharp . . . . . . . Inequalities in . . . . . . . . . . . . . . . . . Alexandrov-Bol’s inequality (1941) projection, and equip it with a metric conformal to the fmat Assume v satisfjes Then . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin . Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Inequality and Determinants . . New Inequality . . . . . . . . . . In general, we can identify a sphere with R 2 by a stereographic metric of R 2 , i.e., ds 2 = e 2 v ( dx 2 1 + dx 2 2 ) . ∆ v + K ( x ) e 2 v = 0 , R 2 with the gaussian curvature K ≤ 1 . ∫ (∫ ∫ e v ds ) 2 ≥ e 2 v )( e 2 v ) ( 4 π − ∂ Ω Ω Ω

New Sharp . . . . . . . . . . . . . . . . . . . . . . . . Outline 1 Lebedev-Milin Inequality and Toeplitz Determinants 2 Aubin-Onofri Inequality 3 Sphere Covering Inequality 4 Logrithemic Determinants . . Inequalities in Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Determinants . New Inequality . . . . . . . . . . . . . 5 New Inequality

0 and M u u New Sharp . . . . . . . . Geometry . . . Inequalities in . . . Logrithemic Determinants and Conformal M 2 2 M M 2 k 0 uds 0 M nds 0 M 1 . F u are fmat. Defjne assume that M M consists of nice boundary with geodesic curvature k 0 , If M M . . . Sphere . . . New Inequality Determinants Logrithemic Inequality Covering Inequality . Aubin-Onofri Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . e u ds 0 . . . . . . . . . . . . . . . . . Given a Riemanian surface ( M , σ 0 ) with Gaussian curvature K 0 and normalized area | M | = 1. Consider a conformal metric on σ = e 2 u on M . If ∂ M = ∅ , defjne ∫ ∫ ∫ F ( u ) = 1 |∇ 0 u | 2 dA 0 + K 0 udA 0 − πχ ( M ) ln( e 2 u dA 0 ) .

New Sharp . . . . . . . . . Inequalities in . . . . . . . . . . . . . Logrithemic Determinants and Conformal Geometry 2 M M M 2 . . . Determinants Changfeng Geometry Analysis and . Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Logrithemic New Inequality Gui . . . . . . . . . . . . . . Lebedev-Milin Given a Riemanian surface ( M , σ 0 ) with Gaussian curvature K 0 and normalized area | M | = 1. Consider a conformal metric on σ = e 2 u on M . If ∂ M = ∅ , defjne ∫ ∫ ∫ F ( u ) = 1 |∇ 0 u | 2 dA 0 + K 0 udA 0 − πχ ( M ) ln( e 2 u dA 0 ) . If ∂ M consists of nice boundary with geodesic curvature k 0 , assume that ( M , σ 0 ) and ( M , σ ) are fmat. Defjne ∫ u ∂ u ∫ ∫ F ( u ) = 1 ∂ nds 0 + k 0 uds 0 − 2 πχ ( M ) ln( e u ds 0 ) . ∂ M ∂ M ∂ M

New Sharp . . Inequalities in . . . . . . . . . . . . . . . . . . . . . Extremals B. Osgood, R. Phillips and P. Sarnak. (1988): Maximizing Det is equivalent to minimizing F . . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality . Covering Inequality Sphere Determinants . . . . New Inequality . . Uniformization, Isospectral Properties, etc. . . . . . . . log Det (∆ σ ) ∂ u ∫ Det (∆ σ 0 ) = − 1 6 π F ( u ) + 1 4 π ∂ nds 0 ∂ M

New Sharp . . . Inequalities in . . . . . . . . . . . . . . . . . . . . . . Extremals B. Osgood, R. Phillips and P. Sarnak. (1988): . . . Inequality Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri . Sphere Covering Inequality Logrithemic . . Determinants . . . . . . . . . New Inequality Uniformization, Isospectral Properties, etc. log Det (∆ σ ) ∂ u ∫ Det (∆ σ 0 ) = − 1 6 π F ( u ) + 1 4 π ∂ nds 0 ∂ M Maximizing log Det (∆ σ ) is equivalent to minimizing F .

New Sharp . . . Inequalities in . . . . . . . . . . . . . . . . . . . . . . Extremals B. Osgood, R. Phillips and P. Sarnak. (1988): . . . Inequality Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri . Sphere Covering Inequality Logrithemic . . Determinants . . . . . . . . . New Inequality Uniformization, Isospectral Properties, etc. log Det (∆ σ ) ∂ u ∫ Det (∆ σ 0 ) = − 1 6 π F ( u ) + 1 4 π ∂ nds 0 ∂ M Maximizing log Det (∆ σ ) is equivalent to minimizing F .

3 with degree at most n S 2 e u p x d H 1 S 2 1 2 u 2 d S 2 ud S 2 e u d Chang-Hang showed: p 0 If all polynomials in n Let 1 (16) 0 there exist If Widom’s observation (1988), Chang-Hang (2019) . . Inequalities in . . . n , then for any R such that N n n S 2 4 u J 1 n n N n 2 4 and and C n 2 N 2 Here, N 1 u C n u N n 1 J . . New Sharp . Logrithemic . . . . . . . . . . New Inequality Determinants Inequality . Covering Sphere Inequality Aubin-Onofri Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . . . . . . . . . . . . . . . . . . . . . ∫ S 1 e ik θ e u d θ = 0 , − n ≤ k ≤ n , then ∫ ∫ log( 1 S 1 e u d θ ) − 1 S 1 ud θ ≤ 4 π ( n + 1 ) ||∇ u || 2 L 2 ( D ) 2 π 2 π

S 2 e u p x d H 1 S 2 1 2 u 2 d S 2 ud S 2 e u d (16) n , then for any p 0 If Let Chang-Hang showed: Widom’s observation (1988), Chang-Hang (2019) 1 If N n . . . Inequalities in . . . 0 there exist J and C n n S 2 4 u J 1 n n N n 2 4 and R such that 2 N 2 Here, N 1 u C n u N n 1 . . New Sharp . Logrithemic . . . . . . . . . . New Inequality Determinants Inequality . Covering Sphere Inequality Aubin-Onofri Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . . . . . . . . . . . . . . . . . . . . ∫ S 1 e ik θ e u d θ = 0 , − n ≤ k ≤ n , then ∫ ∫ log( 1 S 1 e u d θ ) − 1 S 1 ud θ ≤ 4 π ( n + 1 ) ||∇ u || 2 L 2 ( D ) 2 π 2 π P n = { all polynomials in R 3 with degree at most n } .

1 2 u 2 d S 2 ud S 2 e u d Widom’s observation (1988), Chang-Hang (2019) . . . . . . Inequalities in . 1 . . . . . . If Let (16) 2 S 2 4 u J 1 n n N n n Chang-Hang showed: 4 and 2 N 2 Here, N 1 1 J If . . New Sharp . Covering . . . . . New Inequality Determinants Logrithemic Inequality Sphere . Inequality Aubin-Onofri Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . . . . . . . . . . . . . . . . . . ∫ S 1 e ik θ e u d θ = 0 , − n ≤ k ≤ n , then ∫ ∫ log( 1 S 1 e u d θ ) − 1 S 1 ud θ ≤ 4 π ( n + 1 ) ||∇ u || 2 L 2 ( D ) 2 π 2 π P n = { all polynomials in R 3 with degree at most n } . ∫ S 2 e u p ( x ) d ω = 0 , ∀ p ∈ P n , then for any ϵ > 0 there exist N ( n ) ∈ Z and C n ( ϵ ) ∈ R such that N ( n ) + ϵ ( u ) ≥ C n ( ϵ ) > −∞ , ∀ u ∈ H 1 ( S 2 ) .

u 2 d S 2 ud S 2 e u d . . . . . . . Inequalities in . . . . . . . . . If S 2 4 u J 1 J Let . Chang-Hang showed: (16) 1 If Widom’s observation (1988), Chang-Hang (2019) . . New Sharp . Aubin-Onofri Logrithemic . Inequality Covering Sphere Inequality Determinants New Inequality Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and Determinants . . . . . . . . . . . . . . . . . . . ∫ S 1 e ik θ e u d θ = 0 , − n ≤ k ≤ n , then ∫ ∫ log( 1 S 1 e u d θ ) − 1 S 1 ud θ ≤ 4 π ( n + 1 ) ||∇ u || 2 L 2 ( D ) 2 π 2 π P n = { all polynomials in R 3 with degree at most n } . ∫ S 2 e u p ( x ) d ω = 0 , ∀ p ∈ P n , then for any ϵ > 0 there exist N ( n ) ∈ Z and C n ( ϵ ) ∈ R such that N ( n ) + ϵ ( u ) ≥ C n ( ϵ ) > −∞ , ∀ u ∈ H 1 ( S 2 ) . 2 ⌋ + 1 ) 2 ≤ N ( n ) ≤ n ( n + 1 ) Here, N ( 1 ) = 2 , N ( 2 ) = 4 and ( ⌊ n

New Sharp . . . . . . . . . . . Inequalities in . . . . . (16) 4 1 J If Let Chang-Hang showed: 1 . If Widom’s observation (1988), Chang-Hang (2019) . . . . . . . Toeplitz Inequality and Lebedev-Milin Gui Changfeng . Geometry Analysis and Aubin-Onofri Inequality Sphere Covering Inequality Logrithemic Determinants New Inequality . . . . . . . . Determinants . . . . . . . ∫ S 1 e ik θ e u d θ = 0 , − n ≤ k ≤ n , then ∫ ∫ log( 1 S 1 e u d θ ) − 1 S 1 ud θ ≤ 4 π ( n + 1 ) ||∇ u || 2 L 2 ( D ) 2 π 2 π P n = { all polynomials in R 3 with degree at most n } . ∫ S 2 e u p ( x ) d ω = 0 , ∀ p ∈ P n , then for any ϵ > 0 there exist N ( n ) ∈ Z and C n ( ϵ ) ∈ R such that N ( n ) + ϵ ( u ) ≥ C n ( ϵ ) > −∞ , ∀ u ∈ H 1 ( S 2 ) . 2 ⌋ + 1 ) 2 ≤ N ( n ) ≤ n ( n + 1 ) Here, N ( 1 ) = 2 , N ( 2 ) = 4 and ( ⌊ n J α ( u ) = α ∫ ∫ ∫ S 2 |∇ u | 2 d ω + S 2 ud ω − log S 2 e u d ω,

New Sharp . . . . . . . . . . . . . . . . . . . . . . . . Outline 1 Lebedev-Milin Inequality and Toeplitz Determinants 2 Aubin-Onofri Inequality 3 Sphere Covering Inequality 4 Logrithemic Determinants . . Inequalities in Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Determinants . New Inequality . . . . . . . . . . . . . 5 New Inequality

u 2 d H 1 S 2 H 1 S 2 is NOT bounded below in H 1 S 2 for New Sharp . . . . . . . . A Variant of Aubin-Onofri Inequality, Alice Chang Inequalities in . . . . . . Theorem (Chang and G., 2019) and G., 2019 (17) But I u 0 u 2 3 we have I In particular, when u 3 S 2 2 3 u I 1 2 , we have For any . . . . Sphere . . . New Inequality Determinants Logrithemic Inequality Covering Inequality . Aubin-Onofri Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . 2 3 . . . . . . . . . . . . . . . . . . Let us consider the following functionals in H 1 ( S 2 ) : ∫ ∫ I α ( u ) = α S 2 |∇ u | 2 d ω + 2 S 2 ud ω ∫ ∫ S 2 e 2 u d ω ) 2 − ∑ − 1 2 log[( ( S 2 e 2 u x i d ω ) 2 ] . i = 1

H 1 S 2 is NOT bounded below in H 1 S 2 for New Sharp Inequalities in . . . . . . . . . . . . . . . . . . . A Variant of Aubin-Onofri Inequality, Alice Chang and G., 2019 3 Theorem (Chang and G., 2019) (17) In particular, when 2 3 we have I u 0 u But I . . . Aubin-Onofri Logrithemic Inequality Covering . Sphere Inequality Determinants New Inequality Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and Determinants 2 3 . . . . . . . . . . . . . . . . . . Let us consider the following functionals in H 1 ( S 2 ) : ∫ ∫ I α ( u ) = α S 2 |∇ u | 2 d ω + 2 S 2 ud ω ∫ ∫ S 2 e 2 u d ω ) 2 − ∑ − 1 2 log[( ( S 2 e 2 u x i d ω ) 2 ] . i = 1 For any α > 1 / 2 , we have ∫ I α ( u ) ≥ ( α − 2 / 3 ) S 2 |∇ u | 2 d ω, ∀ u ∈ H 1 ( S 2 ) .

is NOT bounded below in H 1 S 2 for New Sharp . . . . . . . . . Inequalities in . . . . . . . . . . . . . A Variant of Aubin-Onofri Inequality, Alice Chang and G., 2019 3 Theorem (Chang and G., 2019) (17) But I . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin . Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality 2 3 . Determinants New Inequality . . . . . . . . . . . . . . Let us consider the following functionals in H 1 ( S 2 ) : ∫ ∫ I α ( u ) = α S 2 |∇ u | 2 d ω + 2 S 2 ud ω ∫ ∫ S 2 e 2 u d ω ) 2 − ∑ − 1 2 log[( ( S 2 e 2 u x i d ω ) 2 ] . i = 1 For any α > 1 / 2 , we have ∫ I α ( u ) ≥ ( α − 2 / 3 ) S 2 |∇ u | 2 d ω, ∀ u ∈ H 1 ( S 2 ) . In particular, when α ≥ 2 / 3 we have I α ( u ) ≥ 0 , ∀ u ∈ H 1 ( S 2 )

New Sharp . . . . . . . . . Inequalities in . . . . . . . . . . . . . . A Variant of Aubin-Onofri Inequality, Alice Chang and G., 2019 3 Theorem (Chang and G., 2019) (17) . . . Logrithemic Analysis and Geometry Changfeng . Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Gui Determinants . . . New Inequality . . . . . . . . . . Let us consider the following functionals in H 1 ( S 2 ) : ∫ ∫ I α ( u ) = α S 2 |∇ u | 2 d ω + 2 S 2 ud ω ∫ ∫ S 2 e 2 u d ω ) 2 − ∑ − 1 2 log[( ( S 2 e 2 u x i d ω ) 2 ] . i = 1 For any α > 1 / 2 , we have ∫ I α ( u ) ≥ ( α − 2 / 3 ) S 2 |∇ u | 2 d ω, ∀ u ∈ H 1 ( S 2 ) . In particular, when α ≥ 2 / 3 we have I α ( u ) ≥ 0 , ∀ u ∈ H 1 ( S 2 ) But I α is NOT bounded below in H 1 ( S 2 ) for α < 2 / 3 .

1 a i x i 1 a 2 . . . . . . . . New Sharp Let . . . . . Inequalities in . Euler-Lagrange Equation Defjne (18) i 0 on S 2 1 e 2 u i i 3 1 3 . 1 u is in The Euler Lagrange equation for the functional I Proposition (19) . . . Inequality . . New Inequality Determinants Logrithemic Inequality Covering Sphere Aubin-Onofri . Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . . (20) . . . . . . . . . . . . . . . . . ∫ a i = S 2 e 2 u x i d ω, i = 1 , 2 , 3 . ∫ H = { u ∈ H 1 ( S 2 ) : S 2 e 2 u d ω = 1 } .

New Sharp . . . . Inequalities in . . . . . . . . . . . . . . . . . . Euler-Lagrange Equation Let (18) Defjne (19) Proposition i . . . Determinants Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants . Inequality Sphere Covering Inequality Logrithemic Aubin-Onofri New Inequality . . . . . . . . . . . . . (20) . ∫ a i = S 2 e 2 u x i d ω, i = 1 , 2 , 3 . ∫ H = { u ∈ H 1 ( S 2 ) : S 2 e 2 u d ω = 1 } . The Euler Lagrange equation for the functional I α in H is α ∆ u + 1 − ∑ 3 e 2 u − 1 = 0 on S 2 . i = 1 a i x i 1 − ∑ 3 i = 1 a 2

a 1 a 2 a 3 New Sharp . . . . . . . . . . . . . . . . . Inequalities in 3 , for any a After a proper rotation, the solution u is explicitly given by the 0 0 0 . In particular, u is axially symmetric about a if a such that (18) holds. solution u to equation (20) in B 1 , there is a unique 2 . ii) When constant solutions; 3 , equation (20) has only Proposition Existence and Nonexistence of Solutions . . . . Determinants Logrithemic Inequality Covering Sphere Inequality Aubin-Onofri Toeplitz . Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and Determinants New Inequality . . . . . . . . . . . . . . . . . formula in (26) below. i ) When α ∈ [ 1 2 , 1 ) and α ̸ = 2

New Sharp . Inequalities in . . . . . . . . . . . . . . . . . . . . . Existence and Nonexistence of Solutions Proposition 3 , equation (20) has only constant solutions; In particular, u is axially symmetric about a if a 0 0 0 . After a proper rotation, the solution u is explicitly given by the . . . Determinants Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering . Logrithemic Inequality New Inequality . . . . . . . formula in (26) below. . . . . . . . i ) When α ∈ [ 1 2 , 1 ) and α ̸ = 2 ii) When α = 2 3 , for any ⃗ a = ( a 1 , a 2 , a 3 ) ∈ B 1 , there is a unique solution u to equation (20) in H such that (18) holds.

New Sharp . . Inequalities in . . . . . . . . . . . . . . . . . . . . . Existence and Nonexistence of Solutions Proposition 3 , equation (20) has only constant solutions; After a proper rotation, the solution u is explicitly given by the . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality . Covering Inequality Sphere Determinants . . . . New Inequality . . formula in (26) below. . . . . . . . i ) When α ∈ [ 1 2 , 1 ) and α ̸ = 2 ii) When α = 2 3 , for any ⃗ a = ( a 1 , a 2 , a 3 ) ∈ B 1 , there is a unique solution u to equation (20) in H such that (18) holds. In particular, u is axially symmetric about ⃗ a if ⃗ a ̸ = ( 0 , 0 , 0 ) .

New Sharp . . Inequalities in . . . . . . . . . . . . . . . . . . . . . Existence and Nonexistence of Solutions Proposition 3 , equation (20) has only constant solutions; After a proper rotation, the solution u is explicitly given by the . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality . Covering Inequality Sphere Determinants . . . . New Inequality . . formula in (26) below. . . . . . . . i ) When α ∈ [ 1 2 , 1 ) and α ̸ = 2 ii) When α = 2 3 , for any ⃗ a = ( a 1 , a 2 , a 3 ) ∈ B 1 , there is a unique solution u to equation (20) in H such that (18) holds. In particular, u is axially symmetric about ⃗ a if ⃗ a ̸ = ( 0 , 0 , 0 ) .

x j e 2 u d New Sharp . . . . . . . . . . . . . . . . Inequalities in . . . . . . Kazdan-Warner condition For the Gaussian curvature equation: (21) we have S 2 K x 0 for each j= 1,2, 3 . . . . Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering Inequality Logrithemic Determinants New Inequality . . . . . . . . . . . . . . (22) ∆ u + K ( x ) e 2 u = 1 on S 2 ,

New Sharp . . . . . . . . . . . . . . . . . . . . . . . . . Kazdan-Warner condition For the Gaussian curvature equation: (21) we have Inequalities in . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri Inequality Sphere Covering . Inequality Determinants . . . . . New Inequality . . . . . . . (22) ∆ u + K ( x ) e 2 u = 1 on S 2 , ∫ S 2 ( ∇ K ( x ) · ∇ x j ) e 2 u d ω = 0 for each j= 1,2, 3 .

1 y y 2 y 2 e 2 w a b 2 y 2 e 2 w dy 2 b 2 (23) 3 u w y 2 . Let be on Use the stereographic projection to transform the equation to New Sharp 2 2 . . . . . . . . Stereographic Project for y 1 1 a 1 0 and 1 b a 1 a where b 2 . (24) 2 0 in 1 6 w Then w satisfjes 2 Inequalities in . . Covering . . . . . . New Inequality Determinants Logrithemic Inequality Sphere . Inequality Aubin-Onofri Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . . . . . . . . . . . . . . (25) . . . . . . . . For α = 2 3 , we assume that ( a 1 , a 2 , a 3 ) = ( 0 , 0 , a ) with a ∈ ( 0 , 1 ) and consider 3 ∆ u + 1 − ax 3 1 − a 2 e 2 u − 1 = 0 on S 2 .

y 2 e 2 w a b 2 y 2 e 2 w dy 2 b 2 New Sharp . Stereographic Project . . . . . . (23) . . . . . Inequalities in . 2 w Use the stereographic projection to transform the equation to a a 1 0 and 1 b a 1 1 Then w satisfjes where b 2 (24) 2 0 in 1 6 . . . . Sphere . . . New Inequality Determinants Logrithemic Inequality Covering Inequality . Aubin-Onofri Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . . . . . . . . . . . . . . . . . . . (25) For α = 2 3 , we assume that ( a 1 , a 2 , a 3 ) = ( 0 , 0 , a ) with a ∈ ( 0 , 1 ) and consider 3 ∆ u + 1 − ax 3 1 − a 2 e 2 u − 1 = 0 on S 2 . be on R 2 . Let w ( y ) := u (Π − 1 ( y )) − 3 2 ln( 1 + | y | 2 ) for y ∈ R 2 .

y 2 e 2 w dy 2 b 2 New Sharp . . . . . . . . . Inequalities in . . . . . . . . . . . Stereographic Project 2 (23) Use the stereographic projection to transform the equation to Then w satisfjes 6 (24) 1 a . . . Determinants Logrithemic Inequality Covering . Inequality Aubin-Onofri Toeplitz New Inequality Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and Determinants Sphere . . . . . . . . . . . (25) . . . . . For α = 2 3 , we assume that ( a 1 , a 2 , a 3 ) = ( 0 , 0 , a ) with a ∈ ( 0 , 1 ) and consider 3 ∆ u + 1 − ax 3 1 − a 2 e 2 u − 1 = 0 on S 2 . be on R 2 . Let w ( y ) := u (Π − 1 ( y )) − 3 2 ln( 1 + | y | 2 ) for y ∈ R 2 . 1 + a ( b 2 + | y | 2 ) e 2 w = 0 in R 2 ∆ w + where b 2 = 1 + a 1 − a > 1 , b > 0 and

New Sharp . . . . . Inequalities in . . . . . . . . . . . . . . . . . Stereographic Project 2 (23) Use the stereographic projection to transform the equation to Then w satisfjes 6 (24) . . . Determinants Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants . Inequality Sphere Covering Inequality Logrithemic Aubin-Onofri New Inequality . . . . . . . . . . . . . (25) . For α = 2 3 , we assume that ( a 1 , a 2 , a 3 ) = ( 0 , 0 , a ) with a ∈ ( 0 , 1 ) and consider 3 ∆ u + 1 − ax 3 1 − a 2 e 2 u − 1 = 0 on S 2 . be on R 2 . Let w ( y ) := u (Π − 1 ( y )) − 3 2 ln( 1 + | y | 2 ) for y ∈ R 2 . 1 + a ( b 2 + | y | 2 ) e 2 w = 0 in R 2 ∆ w + where b 2 = 1 + a 1 − a > 1 , b > 0 and ∫ R 2 ( b 2 + | y | 2 ) e 2 w dy = ( 1 + a ) π.

New Sharp . . . Inequalities in . . . . . . . . . . . . . . . . . . . . . Exact Solution Now it is easy to verify directly that 2 2 . . . Logrithemic Analysis and Geometry Changfeng Gui Lebedev-Milin Inequality and Toeplitz Determinants Aubin-Onofri . Sphere Covering Inequality Inequality Determinants . . . New Inequality . . is a solution to (23). . . . . . . . 2 ln( b 2 + | y | 2 ) + 2 ln b + 1 w ( y ) = − 3 2 ln 1 + b 2 is a solution to (24) and (25), and hence u ( x ) defjned by 2 ln 1 + | y | 2 u ( x ) = u (Π − 1 ( y )) := 3 b 2 + | y | 2 + 2 ln b + 1 2 ln 1 + b 2 (26)

I 2 3 u 2 3 b a 2 a x New Sharp I b Direct computations show that (27) 2 Defjne know that the solution above is a unique solution. Use symmetry result of G.-Moradifam (2018) and uniqueness result of C.S. Lin (2000) on axially symmetric solutions, we b Symmetry and Uniqueness of Solutions . . . . . . . u b if b x 1 a x 1 1 u Indeed, 0 0 2 3 2 if b u I . 3 Inequalities in . . Inequality . . . . . . . New Inequality Determinants Logrithemic Covering . Sphere Inequality Aubin-Onofri Determinants Toeplitz Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and . . . . . . . . . . . . . . . . . . . . . . S 2 α ln 1 + | y | 2 u α, b ( x ) = u α, b (Π − 1 ( y )) := 1 b 2 + | y | 2 + 2 ln b + 1 2 ln 1 + b 2

New Sharp . . . . . . . Inequalities in . . . . . . . . . Defjne Indeed, 3 3 Direct computations show that (27) 2 know that the solution above is a unique solution. . result of C.S. Lin (2000) on axially symmetric solutions, we Use symmetry result of G.-Moradifam (2018) and uniqueness Symmetry and Uniqueness of Solutions . . . . . . . Logrithemic Inequality Covering Sphere Inequality Aubin-Onofri Toeplitz New Inequality Inequality and Lebedev-Milin Gui Changfeng Geometry Analysis and Determinants Determinants . . . . . . . . . . . . . . . . α ln 1 + | y | 2 u α, b ( x ) = u α, b (Π − 1 ( y )) := 1 b 2 + | y | 2 + 2 ln b + 1 2 ln 1 + b 2 lim b →∞ I α ( u α, b ) = −∞ , if α < 2 lim b →∞ I α ( u α, b ) = ∞ , if α > 2 ∀ b > 0 3 ( u 2 3 , b ) = 0 , I 2 a ( x ) = − 1 α ln ( 1 − ⃗ a · x ) + ln( 1 − | ⃗ a | 2 ) , x ∈ S 2 . u α,⃗