The p-laplacian and geometric structure of the Riemannian manifold

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Workshop on Partial Differential Equation and its Applications Institute for Mathematical Sciences - National University of Singapore The p-laplacian and geometric structure of the Riemannian manifold Nguyen Thac Dung (VIASM) Vietnam Institute for Advanced Study in Mathematics December 9, 2014

Outline 1 Analysis on manifolds via the p-laplacian 2

Let (M, g) be a Riemannian manifold with metric tensor g = g ij dx i dx j. The Laplacian on M is defined by ( ) 1 := (det g ij )g ij det gij x i x j where (g ij ) t = (g ij ) 1. It is well-known that 1 is a self-adjoint operator. 2 0 < λ 1 λ 2 <... < λ k <..., lim k λ k =, where λ k is the k-th eigenvalue of φ k = λ k φ k for some 0 φ C (M).

Let (M, g) be a Riemannian manifold with metric tensor g = g ij dx i dx j. The Laplacian on M is defined by ( ) 1 := (det g ij )g ij det gij x i x j where (g ij ) t = (g ij ) 1. It is well-known that 1 is a self-adjoint operator. 2 0 < λ 1 λ 2 <... < λ k <..., lim k λ k =, where λ k is the k-th eigenvalue of φ k = λ k φ k for some 0 φ C (M).

A panoramic view of Riemannian Geometry - Berger Already in the middle ages bell makers knew how to detect invisible cracks by sounding a bell on the ground before lifting it up to the belfry. How can one test the resistance to vibrations of large modern structures by nondestructive essays?... A small crack will not only change the boundary shape of our domain, one side of the crack will strike the other during vibrations invalidating our use of the simple linear wave equation Schuster 1882 Can we hear the shape of a drum? Answer: No by Gordon, Web and Wolpert (1982).

A panoramic view of Riemannian Geometry - Berger Already in the middle ages bell makers knew how to detect invisible cracks by sounding a bell on the ground before lifting it up to the belfry. How can one test the resistance to vibrations of large modern structures by nondestructive essays?... A small crack will not only change the boundary shape of our domain, one side of the crack will strike the other during vibrations invalidating our use of the simple linear wave equation Schuster 1882 Can we hear the shape of a drum? Answer: No by Gordon, Web and Wolpert (1982).

Let Ω be a polygon in R n Mark Kac - 1966 e λkt 1 ( area(ω) 4πtlength(Ω) + 2πt ) (1 γ(ω)), 4πt 3 k=1 where γ(ω) is the genus of Ω. 1967, McKean and Singer: Riemannian manifolds with boundary. 1953, Minakshisundaram: compact Riemmanian manifold M without boundary, ( e λkt 1 V ol(m) t ) R (4πt) n/2 g dv g + O(t 2 ), 6 M k=1 where R g denotes scalar curvature.

Let Ω be a polygon in R n Mark Kac - 1966 e λkt 1 ( area(ω) 4πtlength(Ω) + 2πt ) (1 γ(ω)), 4πt 3 k=1 where γ(ω) is the genus of Ω. 1967, McKean and Singer: Riemannian manifolds with boundary. 1953, Minakshisundaram: compact Riemmanian manifold M without boundary, ( e λkt 1 V ol(m) t ) R (4πt) n/2 g dv g + O(t 2 ), 6 M k=1 where R g denotes scalar curvature.

Question: How about the shape of a complete non-compact Riemannian manifold?

Let M be a complete Riemannian manifold. λ 1 (M): the lower bound of the spectrum of the Laplacian on M } { M λ 1 (M) = inf M φ 2 : φ C φ2 0 (M). Informations on λ 1 (M) = geometry of the manifold. 1 Analytic properties implies geometric properties. 2 Geometric structures of complete Riemannian manifolds.

Let M be a complete Riemannian manifold. λ 1 (M): the lower bound of the spectrum of the Laplacian on M } { M λ 1 (M) = inf M φ 2 : φ C φ2 0 (M). Informations on λ 1 (M) = geometry of the manifold. 1 Analytic properties implies geometric properties. 2 Geometric structures of complete Riemannian manifolds.

Li-Wang: Jour. Diff. Geom., 2002 Let M be an m-dimensional complete Riemannian manifold with n 4. Suppose that Ric M (n 1) and Then either: 1 M has only one end; or λ 1 (M) (n 1)2. 4 2 M = R N with warped product metric ds 2 M = dt 2 + e 2t ds 2 N where N is a compact manifold with nonnegative Ricci curvature.

Li-Wang: Jour. Diff. Geom., 2002 Let M be an m-dimensional complete Riemannian manifold with n 4. Suppose that Ric M (n 1) and Then either: 1 M has only one end; or λ 1 (M) (n 1)2. 4 2 M = R N with warped product metric ds 2 M = dt 2 + e 2t ds 2 N where N is a compact manifold with nonnegative Ricci curvature.

For p > 1, the p-laplacian is defined by p f := div( f p 2 f). Let λ 1 ( p ) be the smallest eigenvalue of p in M. { M λ 1 ( p ) = inf M u p, u u p } C0 (M). 1 Moser (J.Eur. M.S. 2007): Inverse mean curvature flow 2 Kotschwar and Ni (2009), X.D. Wang and L. Zhang (2011): Gradient estimate 3 Y. Wang and J.Y. Chen (2014, preprint): Steady gradient Ricci soliton.

For p > 1, the p-laplacian is defined by p f := div( f p 2 f). Let λ 1 ( p ) be the smallest eigenvalue of p in M. { M λ 1 ( p ) = inf M u p, u u p } C0 (M). 1 Moser (J.Eur. M.S. 2007): Inverse mean curvature flow 2 Kotschwar and Ni (2009), X.D. Wang and L. Zhang (2011): Gradient estimate 3 Y. Wang and J.Y. Chen (2014, preprint): Steady gradient Ricci soliton.

Theorem (Sung-Wang, MRL 2014) Let (M n, g) be a complete manifold of dimension n 2. Suppose that Ric M (n 1) and ( ) p n 1 λ 1 ( p ) =. p Then either 1 M has no finite volume ends; or 2 M = R N with warped product metric ds 2 M = dt 2 + e 2t ds 2 N and N being a compact manifold with nonnegative Ricci curvature.

M. Batista, M. P. Cavalcante, N. L. Santos (Geo. Dedi. 2014) 1 M 2m : complete non-compact Kähler manifold with holomorphic bisectional curvature BK M 1 then λ 1 ( p ) 4p m p p p. 2 M 4m : complete non-compact quaternionic Kähler manifold with scaltar curvature S M 16m(m + 2) then λ 1 ( p ) 2p (2m + 1) p p p.

Theorem 1 Let M 2m be a complete Kähler manifold of complex dimension m 1 with holomorphic bisectional curvature bounded by ( ) p, 2m If λ 1,p p then either 1 M has no p-parabolic end; or BK M 1. 2 M splits as a warped product M = R N where N is a compact manifold. Moreover, the metric is given by 2m ds 2 M = dt 2 + e 4t ω2 2 + e 2t ωα, 2 α=3 where {ω 2,..., ω 2m } are orthonormal coframes for N.

Theorem 2 Let M 4m be a complete noncompact quaternionic Kähler manifold of real dimension 4m with the scalar curvature of M bounded by If λ 1,p ( ) 2(2m+1) p, p then either 1 M has no p-parabolic end; or S M 16m(m + 2). 2 M splits as a warped product M = R N where N is a compact manifold. Moreover, the metric is given by ds 2 M = dt 2 + e 4t 4 4m ωp 2 + e 2t ωα, 2 p=2 α=5 where {ω 2,..., ω 4m } are orthonormal coframes for N.

Let E M be an end. E is p-parabolic if it does not admit a p-harmonic function f : E R such that { f E = 1; lim inf f(y) < 1. y E,y Remark Li-Wang (Amer. Jour. Math, 2009) proved the Theorem 1, when p = 2 Kong-Li-Zhou (JDG, 2008) proved the Theorem 2, when p = 2.

Suppose that M has a p-parabolic end E. Let β be the Busemann function associated with a geodesic ray γ contained in E, namely, β(q) = lim t (t dist(q, γ(t))). The Laplacian comparison theorems (by LW, KLZ) imply β a, where a = 2m in theorem 1 (Li-Wang) and a = 2(2m + 1) in theorem 2 (Kong-Li-Zhou). Claim β = a Idea: for b = a p, let g = ebβ, we can show p (g) λ 1,p g p 1. Using the volume estimate of S. Buckley and P. Koskela (Matth. Zeits., 2005), it turns out that p (g) = λ 1,p g p 1.

Suppose that M has a p-parabolic end E. Let β be the Busemann function associated with a geodesic ray γ contained in E, namely, β(q) = lim t (t dist(q, γ(t))). The Laplacian comparison theorems (by LW, KLZ) imply β a, where a = 2m in theorem 1 (Li-Wang) and a = 2(2m + 1) in theorem 2 (Kong-Li-Zhou). Claim β = a Idea: for b = a p, let g = ebβ, we can show p (g) λ 1,p g p 1. Using the volume estimate of S. Buckley and P. Koskela (Matth. Zeits., 2005), it turns out that p (g) = λ 1,p g p 1.

A smooth metric measure space (M, g, e f dv) contains a Riemannian manifold: (M, g) a weighted volume form: e f dv, where f C (M). dv is the volume element induced by the Riemannian metric g.

For p > 1, a function u is said to be weighted p-harmonic if f,p u := e f div(e f u p 2 u) = 0. Note that weighted p-harmonic functions are characterized as the critical points of the weighted Dirichlet energy u p e f dv. M If u is a weighted p-harmonic function and u p e f dv < + M then u is said to have finite p-energy. The bottom spectrum of the weighted Laplacian f,p, { } M λ 1,p := inf φ p e f dv M φp e f dv, φ C 0 (M).

For p > 1, a function u is said to be weighted p-harmonic if f,p u := e f div(e f u p 2 u) = 0. Note that weighted p-harmonic functions are characterized as the critical points of the weighted Dirichlet energy u p e f dv. M If u is a weighted p-harmonic function and u p e f dv < + M then u is said to have finite p-energy. The bottom spectrum of the weighted Laplacian f,p, { } M λ 1,p := inf φ p e f dv M φp e f dv, φ C 0 (M).

For p > 1, a function u is said to be weighted p-harmonic if f,p u := e f div(e f u p 2 u) = 0. Note that weighted p-harmonic functions are characterized as the critical points of the weighted Dirichlet energy u p e f dv. M If u is a weighted p-harmonic function and u p e f dv < + M then u is said to have finite p-energy. The bottom spectrum of the weighted Laplacian f,p, { } M λ 1,p := inf φ p e f dv M φp e f dv, φ C 0 (M).

For any m N, the m dimensional Bakry-Émery curvature Ricm f associated to smooth metric measure space (M, g, e f dv) is defined by where Ric m f = Ric + Hess(f) f f m n, Ric denotes the Ricci curvature of (M, g), and Hess(f) the Hessian of f. m n. m = n if and only if f is constant. We also define Ric f := Ric + Hess(f)

Proposition (Y. Wang and J. Y. Chen, 2014) Let g = e 2f n p be a conformal change of g. Then u is a weighted p-harmonic function on (M, g) if and only if u is a p-harmonic function on (M, g).

Theorem 3 (Local gradient estimate) Let (M n, g, e f ) be a smooth metric measure space of dimension n with Ric m f (m 1)κ. Suppose that v is a positive smooth weighted p-harmonic function on the ball B R = B(o, R) M. Then there exists a constant C = C(p, m, n) such that v v C(1 + κr) R on B(o, R/2).

Ideas of proof. 1 Laplacian comparison f ρ := f, ρ (m 1) κ coth( κρ) provided that Ric m f (m 1)κ. Here ρ(x) := dist(o, x) stands for the distance between x M and a fixed point o M. 2 Volume comparison 3 Sobolev inequality. V f (B x (r 2 )) V f (B x (r 1 )) V Hm(r 2 ) V H m(r 1 )

Sobolev inequality Let (M, g, e f dµ) be an n-dimensional complete noncompact smooth metric measure space. If Ric m f (m 1)κ for some nonnegative constants κ, then for any p > 1, there exists a constant c = c(n, p, m) > 0 depending only on p, n, m such that ( B R ϕ 2p p 2 e f ) p 2 p R2.e c(1+ κr) ( ϕ 2 + R 2 ϕ 2) e f V f (B R ) 2 p B R for any ϕ C 0 (B p (R)).

By theorem 3, v v C(1 + κr) R on B(o, R/2). Harnack inequality Let (M, g, e f dv) be a complete smms of dimension n 2 with Ric m f (m 1)κ. Suppose that v is a positive weighted p-harmonic function on the geodesic ball B(o, R) M. There exists a constant C p,n,m depending only on p, n, m such that v(x) e Cp,n,m(1+ κr) v(y), x, y B(o, R/2). If κ = 0, we have a uniform constant c p,n,m (independent of R) such that sup B(o,R/2) v c p,n inf v. B(o,R/2)

Liouville property Assume that (M, g) is a smms with Ric m f 0. If u is a weighted p-harmonic function bounded from below on M then u is constant. Estimate of the first weighted eigenvalue Let (M n, g, e f dv) be an n-dimensional complete noncompact manifold with Ric m f (m 1). Then ( ) p m 1 λ 1,p. p 1 Volume estimate of Buckley and Koskela, 2 Volume comparison theorem V f (B(r)) Ce pλ1/p 1,p r, r large enough V f (B(r)) C 1 e (m 1)r.

Liouville property Assume that (M, g) is a smms with Ric m f 0. If u is a weighted p-harmonic function bounded from below on M then u is constant. Estimate of the first weighted eigenvalue Let (M n, g, e f dv) be an n-dimensional complete noncompact manifold with Ric m f (m 1). Then ( ) p m 1 λ 1,p. p 1 Volume estimate of Buckley and Koskela, 2 Volume comparison theorem V f (B(r)) Ce pλ1/p 1,p r, r large enough V f (B(r)) C 1 e (m 1)r.

Liouville property Assume that (M, g) is a smms with Ric m f 0. If u is a weighted p-harmonic function bounded from below on M then u is constant. Estimate of the first weighted eigenvalue Let (M n, g, e f dv) be an n-dimensional complete noncompact manifold with Ric m f (m 1). Then ( ) p m 1 λ 1,p. p 1 Volume estimate of Buckley and Koskela, 2 Volume comparison theorem V f (B(r)) Ce pλ1/p 1,p r, r large enough V f (B(r)) C 1 e (m 1)r.

Theorem 4 (Global sharp gradient estimate) Let (M n, g, e f dv) be an n-dimensional complete noncompact manifold with Ric m f (m 1). If v is a positive weighted p-eigenfunction with respect to the first eigenvalue λ 1,p, that is, then e f div(e f v p 2 v) = λ 1,p v p 1 ln v y. Here y is the unique positive root of the equation Corollary (p 1)y p (m 1)y p 1 + λ 1,p = 0. Let (M n, g, e f dv) be an n-dimensional complete noncompact manifold with Ric m f (m 1). If v is a positive weighted p-harmonic function then ln v m 1 p 1.

Theorem 4 (Global sharp gradient estimate) Let (M n, g, e f dv) be an n-dimensional complete noncompact manifold with Ric m f (m 1). If v is a positive weighted p-eigenfunction with respect to the first eigenvalue λ 1,p, that is, then e f div(e f v p 2 v) = λ 1,p v p 1 ln v y. Here y is the unique positive root of the equation Corollary (p 1)y p (m 1)y p 1 + λ 1,p = 0. Let (M n, g, e f dv) be an n-dimensional complete noncompact manifold with Ric m f (m 1). If v is a positive weighted p-harmonic function then ln v m 1 p 1.

Let M n = R N n 1 with a warped product metric ds 2 = dt 2 + e 2t ds 2 N, where N is a complete manifold with non-negative Ricci curvature. Then we can compute Ric M (n 1) = 2 t 2 + (n 1) t + e 2t N. Choose weighted function f = (m n)t, then Ric m f (m 1). Let v(t, x) = e at, where m 1 p a m 1 p 1, we have v v = a. e f div(e f v p 2 v) = ((p 1)a (m 1))a p 1 v p 1. This implies that λ 1,p = (m 1 (p 1)a)a p 1, or (p 1)a p (m 1)a p 1 + λ 1,p = 0.

Let M n = R N n 1 with a warped product metric ds 2 = dt 2 + e 2t ds 2 N, where N is a complete manifold with non-negative Ricci curvature. Then we can compute Ric M (n 1) = 2 t 2 + (n 1) t + e 2t N. Choose weighted function f = (m n)t, then Ric m f (m 1). Let v(t, x) = e at, where m 1 p a m 1 p 1, we have v v = a. e f div(e f v p 2 v) = ((p 1)a (m 1))a p 1 v p 1. This implies that λ 1,p = (m 1 (p 1)a)a p 1, or (p 1)a p (m 1)a p 1 + λ 1,p = 0.

Theorem 5 Let (M n, g, e f dv) be a smooth metric measure space ( of dimension ) p. n 2. Suppose that Ric m f (m 1) and λ 1,p = Then m 1 p either M has no p-parabolic ends or M = R N n 1 for some compact manifold N. Suppose that M has a p-parabolic end E. Let β be the Busemann function associated with a geodesic ray γ containd in E, namely, β(x) = lim t (t dist(x, γ(t))). 1 Laplacian comparison theorem, 2 let ω := e m 1 p β, we obtain f β (m 1). p,f (ω) λ 1,p ω p 1.

Theorem 5 Let (M n, g, e f dv) be a smooth metric measure space ( of dimension ) p. n 2. Suppose that Ric m f (m 1) and λ 1,p = Then m 1 p either M has no p-parabolic ends or M = R N n 1 for some compact manifold N. Suppose that M has a p-parabolic end E. Let β be the Busemann function associated with a geodesic ray γ containd in E, namely, β(x) = lim t (t dist(x, γ(t))). 1 Laplacian comparison theorem, 2 let ω := e m 1 p β, we obtain f β (m 1). p,f (ω) λ 1,p ω p 1.

Theorem 6 (Rigidity - D) Let (M n, g, e f dv) be a smooth metric measure space ( of dimension ) p n 3. Suppose that Ric m f (m 1) and λ 1,p = for some m 1 p 2 p (m 1)2 2(m 2). Then either M has only one p-nonparabolic end or M = R N n 1 for some compact manifold N. An open and interesting problem Can we get the same results if Ric f (m 1).

Theorem 6 (Rigidity - D) Let (M n, g, e f dv) be a smooth metric measure space ( of dimension ) p n 3. Suppose that Ric m f (m 1) and λ 1,p = for some m 1 p 2 p (m 1)2 2(m 2). Then either M has only one p-nonparabolic end or M = R N n 1 for some compact manifold N. An open and interesting problem Can we get the same results if Ric f (m 1).

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