Lower bounds for the first eigenvalue of $p$ -Laplacian on Kähler manifolds

Kui Wang School of Mathematical Sciences, Soochow University, Suzhou, 215006, China [email protected] and Shaoheng Zhang School of Mathematical Sciences, Soochow University, Suzhou, 215006, China [email protected]

Abstract.

We study the eigenvalue problem for the $p$ -Laplacian on Kähler manifolds. Our first result is a lower bound for the first nonzero eigenvalue of the $p$ -Laplacian on compact Kähler manifolds in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature for $p\in(1,2]$ . Our second result is a sharp lower bound for the first Dirichlet eigenvalue of the $p$ -Laplacian on compact Kähler manifolds with smooth boundary for $p\in(1,\infty)$ . Our results generalize corresponding results for the Laplace eigenvalues on Kähler manifolds proved in [14].

Key words and phrases:

Eigenvalue of

p

-Laplacian, modulus of continuity, Kähler manifolds

2010 Mathematics Subject Classification:

35P15, 53C55

The research of the first author is supported by NSFC No.11601359

1. Introduction and Main Results

Let $(M^{m},g)$ be an $m$ -dimensional compact Riemannian manifold possibly with smooth boundary and the $p$ -Laplacian operator of the metric $g$ is defined by

\Delta_{p}u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)

for $p\in(1,\infty)$ . The $p$ -Laplacian eigenvalue equation is

(1.1)

-\Delta_{p}u(x)=\lambda|u(x)|^{p-2}u(x),\quad x\in M,

and $\lambda\in\mathbb{R}$ is called a closed eigenvalue when $\partial M=\emptyset$ , a Dirichlet eigenvalue when $\partial M\neq\emptyset$ and $u(x)=0$ on $\partial M$ , a Neumann eigenvalue when $\partial M\neq\emptyset$ and $\partial_{\nu}u(x)=0$ on $\partial M$ . Where $\nu$ denotes the outer unit normal to $\partial M$ .

The purpose of the present paper is to study the first nonzero eigenvalues of the $p$ -Laplacian on compact Kähler manifolds. In a recent paper [14], Li and the first author obtained a lower bound for the first nonzero closed and Neumann eigenvalue (when $\partial M\neq\emptyset$ ) of the Laplacian on compact Kähler manifolds in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature. Precisely, they proved the following theorem.

Theorem 1.1 ([14]).

Let $\left(M^{m},g,J\right)$ be a compact Kähler manifold of complex dimension $m$ and diameter $D$ , whose holomorphic sectional curvature is bounded from below by $4\kappa_{1}$ and orthogonal Ricci curvature is bounded from below by $2(m-1)\kappa_{2}$ for some $\kappa_{1},\kappa_{2}\in\mathbb{R}$ . Let $\mu_{1,2}(M)$ be the first nonzero eigenvalue of the Laplacian on $M$ (with Neumann boundary condition if $M$ has a strictly convex boundary). Then

\mu_{1,2}(M)\geq\bar{\mu}_{1}\left(m,\kappa_{1},\kappa_{2},D\right),

where $\bar{\mu}_{1}\left(m,\kappa_{1},\kappa_{2},D\right)$ is the first Neumann eigenvalue of the one-dimensional eigenvalue problem

(1.2)

\displaystyle\varphi^{\prime\prime}-\left(2(m-1)T_{\kappa_{2}}+T_{4\kappa_{1}}\right)\varphi^{\prime}=-\lambda\varphi

on $[-D/2,D/2]$ . Here and in the rest of this paper, we denote the function $T_{\kappa}$ for $\kappa\in\mathbb{R}$ by

(1.3)

T_{\kappa}(t)=\begin{cases}\sqrt{\kappa}\tan(\sqrt{\kappa}t),&\kappa>0,\\ 0,&\kappa=0,\\ -\sqrt{-\kappa}\tanh(\sqrt{-\kappa}t),&\kappa<0.\end{cases}

On Riemannian manifolds, the above theorem known as Zhong-Yang estimate is proved by Zhong and Yang [27] for the case $\operatorname{Ric}\geq 0$ , and by Kröger [9] for the case $\operatorname{Ric}\geq(n-1)\kappa$ for general $\kappa\in\mathbb{R}$ . These works use the gradient estimates method initiated by Li [10] and Li-Yau [12]. In 2013, Andrews and Clutterbuck [4] gave a simple proof via the modulus of continuity estimates for the solutions to the heat equation, see also [6] for a coupling method, and [26] for an elliptic proof based on [1] and [20]. Recently, Valtorta [25] and Naber-Valtorta [19] established the Zhong-Yang estimate for the first nonzero eigenvalue for $p$ -Laplacian on Riemannian manifolds.

On compact Kähler manifolds, Li and the first author proved a lower bound of the first nonzero eigenvalue of Laplacian in terms of geometric data (see Theorem 1.1), and Rutkowski and Seto [22] established an explicit lower bound by studying the one-dimensional ODE (1.2). On noncompact Kähler manifolds, Li-Wang [11] and Munteanu [18] obtained upper bounds of the first Laplace eigenvalue under conditions of bisectional curvature and Ricci curvature respectively. On closed Kähler manifolds with positive Ricci curvature bound from below, Lichnerowicz [17] obtained a optimal lower bound of the first nonzero eigenvalue of Laplacian by using Reilly formula, and generalized by Blacker and Seto [5] to the first nonzero eigenvalue of $p$ -Laplacian for $p\geq 2$ . It is thus a natural question to study the first nonzero eigenvalue of $p$ -Laplacian on Kähler manifolds for $p\in(1,2]$ . Regarding this, we prove

Theorem 1.2.

Let $(M^{m},g,J)$ be a compact Kähler manifold of complex dimension $m$ and diameter $D$ , whose holomorphic sectional curvature is bounded from below by $4\kappa_{1}$ and orthogonal Ricci curvature is bounded from below by $2(m-1)\kappa_{2}$ for some $\kappa_{1},\kappa_{2}\in\mathbb{R}$ . Let $\mu_{1,p}(M)$ be the first nonzero eigenvalue of the $p$ -Laplacian on M (with Neumann boundary condition if $M$ has a strictly convex boundary). Assume $1<p\leq 2$ , then

(1.4)

\displaystyle\mu_{1,p}(M)\geq\bar{\mu}_{1}(m,p,\kappa_{1},\kappa_{2},D),

where $\bar{\mu}_{1}(m,p,\kappa_{1},\kappa_{2},D)$ is the first nonzero Neumann eigenvalue of the one-dimensional eigenvalue problem

(1.5)

(p-1)|\varphi^{\prime}|^{p-2}\varphi^{\prime\prime}-(2(m-1)T_{\kappa_{2}}+T_{4\kappa_{1}})|\varphi^{\prime}|^{p-2}\varphi^{\prime}=-\mu|\varphi|^{p-2}\varphi

on interval $[-D/2,D/2]$ .

Note that if $\kappa_{1}=\kappa_{2}=0$ , ODE (1.5) can be solved by $p$ -trigonometric functions, then as a direct consequence of Theorem 1.2, we have

Corollary 1.3.

With the same assumptions as in Theorem 1.2, and assume $\kappa_{1}=\kappa_{2}=0$ . Then

(1.6)

\mu_{1,p}(M)\geq(p-1)(\frac{\pi_{p}}{D})^{p},

where $\displaystyle\pi_{p}=\frac{2\pi}{p\sin(\pi/p)}$ .

Now let us turn to lower bounds for Dirichlet eigenvalues on compact Kähler manifolds with smooth boundary. For $\kappa,\Lambda\in\mathbb{R}$ , denote by $C_{\kappa,\Lambda}$ the unique solution of the initial value problem

(1.7)

\left\{\begin{aligned} &\phi^{\prime\prime}(t)+\kappa\phi(t)=0,\\ &\phi(0)=1,\ \phi^{\prime}(0)=-\Lambda,\end{aligned}\right.

and denote $T_{\kappa,\Lambda}$ for $\kappa,\Lambda\in\mathbb{R}$ by

(1.8)

T_{\kappa,\Lambda}(t):=-\frac{C_{\kappa,\Lambda}^{\prime}(t)}{C_{\kappa,\Lambda}(t)}.

For the first Dirichlet eigenvalue on compact Kähler manifolds, Li and the first author [14, Theorem 1.4] obtained a lower bound of the first Dirichlet eigenvalue of Laplacian using Barta’s inequality, and Blacker-Seto [5] proved a lower bound of the first Dirichlet eigenvalue of $p$ -Laplacian for $p\geq 2$ via the $p$ -Reilly formula. Our second main result is the following $p$ -Laplacian analogue of the lower bounds for the first Dirichlet eigenvalue on Kähler manifolds for $1<p<\infty$ .

Theorem 1.4.

Let $(M^{m},g,J)$ be a compact Kähler manifold with smooth boundary $\partial M$ . Suppose that the holomorphic sectional curvature is bounded from below by $4\kappa_{1}$ and the orthogonal Ricci curvature is bounded from below by $2(m-1)\kappa_{2}$ for $\kappa_{1},\kappa_{2}\in\mathbb{R}$ , and the second fundamental form on $\partial M$ is bounded from below by $\Lambda\in\mathbb{R}$ . Let $\lambda_{1,p}(M)$ be the first Dirichlet eigenvalue of the $p$ -Laplacian on $M$ . Assume $1<p<\infty$ , then

(1.9)

\displaystyle\lambda_{1,p}(M)\geq\bar{\lambda}_{1}(m,p,\kappa_{1},\kappa_{2},D),

where $\bar{\lambda}_{1}(m,p,\kappa_{1},\kappa_{2},D)$ is the first eigenvalue of the one-dimensional eigenvalue problem

(1.10)

\displaystyle(p-1)|\varphi^{\prime}|^{p-2}\varphi^{\prime\prime}-\big{(}2(m-1)T_{\kappa_{2},\Lambda}+T_{4\kappa_{1},\Lambda}\big{)}|\varphi^{\prime}|^{p-2}\varphi^{\prime}=-\lambda|\varphi|^{p-2}\varphi

on $[0,D/2]$ with boundary conditions $\varphi(0)=0$ and $\varphi^{\prime}(R)=0$ .

The proof of Theorem 1.4 relies on a comparison theorem for the second derivatives of distance to the boundary proved in [14, Section 6] and Barta’s inequality. In the Riemannian case, lower bounds for the first Dirichlet eigenvalue were proved by Li-Yau [12] and by Kause [8]. It remains an interesting question that whether or not the same result as Theorem 1.2 holds for $p>2$ and the result is sharp. We plan to return to this in the future. The rest of the paper is organized as follows. In Section 2, we recall the definitions of curvatures of Kähler manifolds and modulus of continuity of real functions. Sections 3 and 4 are devoted to proving Theorem 1.2 and Theorem 1.4.

2. Preliminary

2.1. Curvatures of Kähler Manifolds

Let $(M^{m},g,J)$ be a Kähler manifold with complex dimension $m$ (real dimension is $n=2m$ ). A plane $\sigma\subset T_{p}M$ is said to be holomorphic if it is invariant by the complex structure tensor $J$ . The restriction of the sectional curvature to holomorphic planes is called the holomorphic sectional curvature, denoted by $H$ . In other words, if $\sigma$ is a holomorphic plane spanned by $X$ and $JX$ , then the holomorphic sectional curvature of $\sigma$ is defined by

H(\sigma):=H(X)=\frac{R(X,JX,X,JX)}{|X|^{4}}.

We say the holomorphic sectional curvature is bounded from below by $\kappa$ , if $H(\sigma)\geq\kappa$ for all holomorphic planes $\sigma\subset T_{p}M$ and all $p\in M$ . The orthogonal Ricci curvature, denoted by $\operatorname{Ric}^{\perp}$ , is defined for any $X\in T_{p}M$ by

\operatorname{Ric}^{\perp}(X,X):=\operatorname{Ric}(X,X)-H(X)|X|^{2}.

We say the orthogonal Ricci curvature is bounded from below by $\kappa\in\mathbb{R}$ , if $\operatorname{Ric}^{\perp}(X,X)\geq\kappa g(X,X),\forall X\in T_{p}M,\forall p\in M$ ,

Remark 2.1.

If $M$ is a complete Kähler manifold with holomorphic sectional curvature satisfies $H\geq\kappa_{1}$ for some positive $\kappa_{1}$ , Tsukamoto [24] proved that the diameter of $M$ is bounded from above by $\pi/\sqrt{\kappa_{1}}$ . If $M$ is a complete Kähler manifold with orthogonal Ricci curvature satisfies $\operatorname{Ric}^{\perp}\geq 2(m-1)\kappa_{2}$ for some positive $\kappa_{2}$ , Ni and Zheng [21] proved that the diameter of $M$ is bounded from above by $\pi/\sqrt{\kappa_{2}}$ .

2.2. Modulus of Continuity

Let $u$ be a continuous function on a metric space $(X,d)$ , define the modulus of continuity $\omega$ of $u$ by

\omega(s):=\sup\big{\{}\frac{u(x)-u(y)}{2}|\ d(x,y)=2s,x,y\in X\big{\}}.

Recently, Andrews and Clutterbuck [1, 2, 3, 4] investigated how the modulus of continuity of solutions to parabolic differential equations evolves, and they proved for a large class of parabolic equation, the modulus of continuity of the solution is a viscosity solution of the associated one-dimensional equations. By using the modulus of continuity, Andrews and Clutterbuck obtained the sharp lower bound on the fundamental gap for Schrödinger operators [3]. This technique is called modulus of continuity estimate, and is widely used to study the lower bound of the first nonzero eigenvalue in terms of geometric data such as the diameter and dimension of the manifold, see [7, 16, 23] and so on.

3. The First Nonzero Eigenvalue

In this section, we shall use the method of modulus of continuity estimates to prove Theorem 1.2. The proof presented below is a modification of the argument outlined in the survey by Andrews [1, Section 8] for the case of Riemannian manifolds. Let $\varphi(s,t)\in C^{2,1}([0,a]\times[0,\infty))$ , and denote by

(3.1)

\displaystyle\mathfrak{L}\varphi:=(p-1)|\varphi^{\prime}|^{p-2}\varphi^{\prime\prime}-(2(m-1)T_{\kappa_{2}}+T_{4\kappa_{1}})|\varphi^{\prime}|^{p-2}\varphi^{\prime},

where we denote $\frac{\partial^{k}\varphi}{\partial s^{k}}$ by $\varphi^{(k)}$ for short. We first prove the following modulus of continuity estimates for solutions to a nonlinear parabolic equation on Kähler manifolds.

Theorem 3.1.

Let $(M^{m},g,J)$ be a compact Kähler manifold with diameter $D$ whose holomorphic sectional curvature is bounded from below by $4\kappa_{1}$ and the orthogonal Ricci curvature is bounded from below by $2(m-1)\kappa_{2}$ for some $\kappa_{1},\kappa_{2}\in\mathbb{R}$ . Let $v:M\times[0,T)\rightarrow\mathbb{R}$ be a solution of

(3.2)

v_{t}=|\Delta_{p}v|^{-\frac{p-2}{p-1}}\Delta_{p}v

(with Neumann boundary condition if $M$ has a strictly convex boundary). Suppose $1<p\leq 2$ and $\varphi(s,t):[0,\frac{D}{2}]\times[0,\infty)\rightarrow\mathbb{R}$ satisfies

(1)

$v(y,0)-v(x,0)\leq 2\varphi(\frac{d(x,y)}{2},0),\ \forall x,y\in M$ ,
(2)

$0\geq\varphi_{t}\geq|\mathfrak{L}\varphi|^{-\frac{p-2}{p-1}}\mathfrak{L}\varphi,\ (s,t)\in[0,D/2]\times\mathbb{R}_{+}$ ,
(3)

$\varphi^{\prime}(s,t)>0,\ (s,t)\in[0,D/2]\times\mathbb{R}_{+}$ ,
(4)

$\varphi(0,t)\geq 0,\ t>0$ .

Then

(3.3)

\displaystyle v(y,t)-v(x,t)\leq 2\varphi(\frac{d(x,y)}{2},t)

for $t>0$ and $x,y\in M$ .

Proof.

For $\varepsilon>0$ , let

\displaystyle A_{\varepsilon}(x,y,t)=v(y,t)-v(x,t)-2\varphi(\frac{d(x,y)}{2},t)-\varepsilon e^{t}.

To prove (3.3), it suffices to show that $A_{\varepsilon}<0$ for any $\varepsilon>0$ .

We argue by contradiction and assume that there exists $(x_{0},y_{0},t_{0})$ such that $A_{\varepsilon}$ attains its maximum zero on $M\times M\times[0,t_{0}]$ at $(x_{0},y_{0},t_{0})$ . Clearly $x_{0}\neq y_{0}$ , and $t_{0}>0$ . If $\partial M\neq\emptyset$ , similarly as in [14, Theorem 3.1], the strictly convexity of boundary, the Neumann condition and the positivity of $\varphi^{\prime}$ rule out the possibility that $x_{0}\in\partial M$ and $y_{0}\in\partial M$ .

Compactness of $M$ implies that there exists an arc-length minimizing geodesic $\gamma_{0}$ connecting $x_{0}$ and $y_{0}$ such that $\gamma_{0}(-s_{0})=x_{0}$ and $\gamma_{0}(s_{0})=y_{0}$ with $s_{0}=d(x_{0},y_{0})/2$ . We pick an orthonormal basis $\{e_{i}(s)\}_{i=1}^{2m}$ for $T_{x_{0}}M$ with $e_{1}=\gamma^{\prime}_{0}(-s_{0})$ and $e_{2}=J\gamma_{0}^{\prime}(-s_{0})$ , where $J$ is the complex structure. Then parallel transport along $\gamma_{0}$ produces an orthonormal basis $\{e_{i}(s)\}_{i=1}^{2m}$ for $T_{\gamma_{0}(s)}M$ with $e_{1}(s)=\gamma_{0}^{\prime}(s)$ for each $s\in[-s_{0},s_{0}]$ . Since $J$ is parallel and $\gamma_{0}$ is a geodesic, we have $e_{2}(s)=J\gamma_{0}^{\prime}(s)$ for each $s\in[-s_{0},s_{0}]$ .

First derivative inequality yields

(3.4)

0\leq\partial_{t}A_{\varepsilon}(x_{0},y_{0},t_{0})=v_{t}(y_{0},t_{0})-v_{t}(x_{0},t_{0})-2\varphi_{t}-\varepsilon e^{t_{0}},

and

\nabla_{x}A_{\varepsilon}(x_{0},y_{0},t_{0})=\nabla_{y}A_{\varepsilon}(x_{0},y_{0},t_{0})=0,

namely

(3.5)

\nabla v(x_{0},t_{0})=\varphi^{\prime}(s_{0},t_{0})e_{1}(-s_{0}),\nabla v(y_{0},t_{0})=\varphi^{\prime}(s_{0},t_{0})e_{1}(s_{0}).

Recall that

(3.6)

\Delta_{p}v=|\nabla v|^{p-2}\Delta v+(p-2)|\nabla v|^{p-4}\nabla^{2}v(\nabla v,\nabla v),

then plugging equality (3.5) into (3.6) we get

(3.7)

\displaystyle\begin{split}&\Delta_{p}v(y_{0},t_{0})-\Delta_{p}v(x_{0},t_{0})\\ =&(p-1)|\varphi^{\prime}|^{p-2}(v_{11}(y_{0},t_{0})-v_{11}(x_{0},t_{0}))+|\varphi^{\prime}|^{p-2}\sum_{i=2}^{2m}(v_{ii}(y_{0},t_{0})-v_{ii}(x_{0},t_{0})).\end{split}

Now we use the first and second variation formulas of arc length to calculate the second derivatives in space variables. Suppose $\gamma(r,s):[-\delta,\delta]\times[-s_{0},s_{0}]\rightarrow M$ is a smooth variation of $\gamma_{0}$ such that $\gamma(0,s)=\gamma_{0}(s)$ , then the variation formulas give

(3.8)

\displaystyle\frac{\mathrm{d}}{\mathrm{d}r}\big{|}_{r=0}L[\gamma(r,s)]=g(T,\gamma_{r})\big{|}_{-s_{0}}^{s_{0}},

and

(3.9)

\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}\big{|}_{r=0}L[\gamma(r,s)]=\int_{-s_{0}}^{s_{0}}(|(\nabla_{s}\gamma_{r})^{\perp}|^{2}-R(T,\gamma_{r},T,\gamma_{r}))\ \mathrm{d}s+g(T,\nabla_{r}\gamma_{r})\big{|}_{-s_{0}}^{s_{0}},

where $T=\partial_{s}\gamma(0,s)$ is the unit tangent vector to $\gamma_{0}$ .

To calculate the second derivative along $e_{1}$ , we consider the variation $\gamma(r,s):=\gamma_{0}(s+r\frac{s}{s_{0}})$ , and $T=\gamma_{0}^{\prime}(s)$ and $\gamma_{r}=\gamma_{0}^{\prime}(s)s/s_{0}$ . Then formulas (3.8) and (3.9) give

\frac{\mathrm{d}}{\mathrm{d}r}\big{|}_{r=0}L[\gamma(r)]=2,\quad\text{and}\quad\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}\big{|}_{r=0}L[\gamma(r)]=0.

Hence the second derivative test for this variation produces

(3.10)

v_{11}(y_{0},t_{0})-v_{11}(x_{0},t_{0})-2\varphi^{\prime\prime}(s_{0},t_{0})\leq 0.

To calculate the second derivative along $e_{2}$ , we denote by

(3.11)

c_{\kappa}(t)=\begin{cases}\operatorname{cos}(\sqrt{\kappa}t),&\kappa>0,\\ 1,&\kappa=0,\\ \operatorname{cosh}(\sqrt{-\kappa}t),&\kappa<0,\end{cases}

and consider the variation

\gamma(r,s):=\operatorname{exp}_{\gamma_{0}(s)}(r\eta(s)e_{2}(s)),

where $\displaystyle\eta(s)=\frac{c_{4\kappa_{1}}(s)}{c_{4\kappa_{1}}(s_{0})}$ . Clearly $T=e_{1}(s)$ and $\gamma_{r}=\eta(s)e_{2}(s)$ , and formulas (3.8) and (3.9) imply

\frac{\mathrm{d}}{\mathrm{d}r}\bigg{|}_{r=0}L[\gamma(r)]=0

and

\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}\bigg{|}_{r=0}L[\gamma(r)]=\int_{-s_{0}}^{s_{0}}\bigg{(}(\eta^{\prime})^{2}-\eta^{2}R(e_{1},e_{2},e_{1},e_{2})\bigg{)}\ \mathrm{d}s.

So this variation produces

v_{22}(y_{0},t_{0})-v_{22}(x_{0},t_{0})-\varphi^{\prime}(s_{0},t_{0})\int_{-s_{0}}^{s_{0}}\bigg{(}(\eta^{\prime})^{2}-\eta^{2}R(e_{1},e_{2},e_{1},e_{2})\bigg{)}\ \mathrm{d}s\leq 0.

Using the assumption that $H\geq 4\kappa_{1}$ and integration by parts, we estimate that

		$\displaystyle\int_{-s_{0}}^{s_{0}}\bigg{(}(\eta^{\prime})^{2}-\eta^{2}R(e_{1},e_{2},e_{1},e_{2})\bigg{)}\ \mathrm{d}s$
	$\displaystyle=$	$\displaystyle\eta^{\prime}\eta\|_{-s_{0}}^{s_{0}}-\int_{-s_{0}}^{s_{0}}\eta^{2}\big{(}R(e_{1},e_{2},e_{1},e_{2})-4\kappa_{1}\big{)}\ \mathrm{d}s$
	$\displaystyle\leq$	$\displaystyle\eta^{\prime}\eta\|_{-s_{0}}^{s_{0}}-\int_{-s_{0}}^{s_{0}}\eta^{2}\big{(}R(e_{1},e_{2},e_{1},e_{2})-4\kappa_{1}\big{)}\ \mathrm{d}s$
	$\displaystyle=$	$\displaystyle-2T_{4\kappa_{1}}(s_{0}).$

Therefore we conclude from the above two inequalities that

(3.12)

v_{22}(y_{0},t_{0})-v_{22}(x_{0},t_{0})\leq-2T_{4\kappa_{1}}(s_{0})\varphi^{\prime}(s_{0},t_{0}),

where we used the assumption that $\varphi^{\prime}(s,t)>0$ .

To calculate the second derivative along $e_{i}$ ( $3\leq i\leq 2m$ ), we consider the variation

\gamma(r,s):=\operatorname{exp}_{\gamma_{0}(s)}(r\zeta(s)e_{i}(s)),

where $\zeta(s)=\frac{c_{\kappa_{2}}(s)}{c_{\kappa_{2}}(s_{0})}$ . Similarly, the second variation formula gives

v_{ii}(y_{0},t_{0})-v_{ii}(x_{0},t_{0})-\varphi^{\prime}(s_{0},t_{0})\int_{-s_{0}}^{s_{0}}\bigg{(}(\zeta^{\prime})^{2}-\zeta^{2}R(e_{1},e_{i},e_{1},e_{i})\bigg{)}\ \mathrm{d}s\leq 0.

Summing over $3\leq i\leq 2m$ yields

\sum_{i=3}^{2m}\big{(}v_{ii}(y_{0},t_{0})-v_{ii}(x_{0},t_{0})\big{)}-\varphi^{\prime}(s_{0},t_{0})\sum_{i=3}^{2m}\int_{-s_{0}}^{s_{0}}\bigg{(}(\zeta^{\prime})^{2}-\zeta^{2}R(e_{1},e_{i},e_{1},e_{i})\bigg{)}\ \mathrm{d}s\leq 0.

Using integration by parts and assumption that $\operatorname{Ric}^{\perp}\geq 2(m-1)\kappa_{2}$ , we have

		$\displaystyle\sum_{i=3}^{2m}\int_{-s_{0}}^{s_{0}}\bigg{(}(\zeta^{\prime})^{2}-\zeta^{2}R(e_{1},e_{i},e_{1},e_{i})\bigg{)}\mathrm{d}s$
	$\displaystyle=$	$\displaystyle-4(m-1)T_{\kappa_{2}}(s_{0})-\int_{-s_{0}}^{s_{0}}\zeta^{2}\big{(}-2(m-1)\kappa_{2}+\sum_{i=3}^{2m}R(e_{1},e_{i},e_{1},e_{i})\big{)}\mathrm{d}s$
	$\displaystyle\leq$	$\displaystyle-4(m-1)T_{\kappa_{2}}(s_{0}),$

thus we get

(3.13)

\sum_{i=3}^{2m}\big{(}v_{ii}(y_{0},t_{0})-v_{ii}(x_{0},t_{0})\big{)}\leq-4(m-1)T_{\kappa_{2}}(s_{0})\varphi^{\prime}(s_{0},t_{0}).

Inserting inequalities (3.10), (3.12) and (3.13) into (3.7), we get

(3.14)

\Delta_{p}v(y_{0},t_{0})-\Delta_{p}v(x_{0},t_{0})\leq 2\mathfrak{L}\varphi(s_{0},t_{0}).

Let $h(t):=|t|^{-\frac{p-2}{p-1}}t$ , which is odd, increasing and convex for all $t>0$ , then (3.14) implies

h(\Delta_{p}v(y_{0},t_{0}))\leq h(\Delta_{p}v(x_{0},t_{0})+2\mathfrak{L}\varphi).

Applying Lemma 2.1 of [15] with $t=\Delta_{p}v(x_{0},t_{0})$ and $\delta=-\mathfrak{L}\varphi\geq 0$ , we get

h(\Delta_{p}v(x_{0},t_{0})+2\mathfrak{L}\varphi)\leq h(\Delta_{p}v(x_{0},t_{0}))+2h(\mathfrak{L}\varphi),

that is

(3.15)

\frac{1}{2}\big{[}|\Delta_{p}v|^{-\frac{p-2}{p-1}}\Delta_{p}v\big{|}_{(y_{0},t_{0})}-|\Delta_{p}v|^{-\frac{p-2}{p-1}}\Delta_{p}v\big{|}_{(x_{0},t_{0})}\big{]}\leq|\mathfrak{L}\varphi|^{-\frac{p-2}{p-1}}\mathfrak{L}\varphi.

Combining (3.4) and (3.15), we get

\varphi_{t}\leq|\mathfrak{L}\varphi|^{-\frac{p-2}{p-1}}\mathfrak{L}\varphi-\frac{\varepsilon}{2}e^{t_{0}}<|\mathfrak{L}\varphi|^{-\frac{p-2}{p-1}}\mathfrak{L}\varphi,

which contradicts the inequality in assumption (2), so

A_{\varepsilon}(x,y,t)<0

holds true for all $x,y\in M$ and $t>0$ . Hence we complete the proof of Theorem 3.1. $\square$

On the interval $[0,D/2]$ , we define the following corresponding one-dimensional eigenvalue problem

\bar{\sigma}_{1}=\inf\bigg{\{}\frac{\int_{0}^{\frac{D}{2}}|\phi^{\prime}|^{p}c_{\kappa_{2}}^{2m-2}c_{4\kappa_{1}}\ \mathrm{d}s}{\int_{0}^{\frac{D}{2}}|\phi|^{p}c_{\kappa_{2}}^{2m-2}c_{4\kappa_{1}}\ \mathrm{d}s}\ \big{|}\phi\in W^{1,p}\big{(}(0,D/2)\big{)}\backslash\{0\},\ \phi(0)=0\bigg{\}},

where $c_{\kappa}(t)$ is defined by (3.11) and $D$ is the diameter of $M$ . Noting from Remark 2.1 that $D\leq\pi/\sqrt{4\kappa_{1}}$ if $\kappa_{1}>0$ , and $D\leq\pi/\sqrt{\kappa_{2}}$ if $\kappa_{2}>0$ .

Lemma 3.1.

\bar{\mu}_{1}(m,p,\kappa_{1},\kappa_{2},D)=\bar{\sigma}_{1}(m,p,\kappa_{1},\kappa_{2},D).

Proof.

Suppose $\phi(s)$ is an eigenfunction on $[0,D/2]$ corresponding to $\bar{\sigma}_{1}(m,p,\kappa_{1},\kappa_{2},D)$ , and define a test function for $\bar{\mu}_{1}(m,p,\kappa_{1},\kappa_{2},D)$ by

\bar{\phi}(s)=\begin{cases}\phi(s),&s\in[0,D/2],\\ -\phi(-s),&s\in[-D/2,0],\end{cases}

then we get

\bar{\mu}_{1}(m,p,\kappa_{1},\kappa_{2},D)\leq\bar{\sigma}_{1}(m,p,\kappa_{1},\kappa_{2},D).

On the other hand, suppose $\psi(s)$ is an eigenfunction corresponding to $\bar{\mu}_{1}(m,p,\kappa_{1},\kappa_{2},D)$ , and then direct calculation gives

(|\psi^{\prime}|^{p-2}\psi^{\prime}c_{\kappa_{2}}^{2m-2}c_{4\kappa_{1}})^{\prime}=-c_{\kappa_{2}}^{2m-2}c_{4\kappa_{1}}\bar{\mu}_{1}|\psi|^{p-2}\psi.

Integrating the above equation over $[-D/2,D/2]$ yields

0=\int_{-\frac{D}{2}}^{\frac{D}{2}}c_{\kappa_{2}}^{2m-2}c_{4\kappa_{1}}|\psi|^{p-2}\psi\ \mathrm{d}s,

so there exists $s_{0}\in(-D/2,D/2)$ such that $\psi(s_{0})=0$ . Without loss of generality, we assume further that $s_{0}\in[0,D/2)$ , and define $\psi(s)=0$ for $s\in[0,s_{0})$ . Then $\psi$ is a test function for $\bar{\sigma}_{1}$ , and we have

	$\displaystyle\bar{\sigma}_{1}(m,p,\kappa_{1},\kappa_{2},D)\leq$	$\displaystyle\frac{\int_{0}^{\frac{D}{2}}\|\psi^{\prime}\|^{p}c_{\kappa_{2}}^{2m-2}c_{4\kappa_{1}}\ \mathrm{d}s}{\int_{0}^{\frac{D}{2}}\|\psi\|^{p}c_{\kappa_{2}}^{2m-2}c_{4\kappa_{1}}\ \mathrm{d}s}$
	$\displaystyle=$	$\displaystyle\frac{\int_{s_{0}}^{\frac{D}{2}}\|\psi^{\prime}\|^{p}c_{\kappa_{2}}^{2m-2}c_{4\kappa_{1}}\ \mathrm{d}s}{\int_{s_{0}}^{\frac{D}{2}}\|\psi\|^{p}c_{\kappa_{2}}^{2m-2}c_{4\kappa_{1}}\ \mathrm{d}s}$
	$\displaystyle=$	$\displaystyle\bar{\mu}_{1}(m,p,\kappa_{1},\kappa_{2},D).$

Thus Lemma 3.1 holds. $\square$

Lemma 3.2.

There exists an odd eigenfunction $\phi$ corresponding to $\bar{\mu}_{1}(m,p,\kappa_{1},\kappa_{2},D)$ satisfying

(3.16)

\displaystyle(p-1)|\phi^{\prime}|^{p-2}\phi^{\prime\prime}-(2(m-1)T_{\kappa_{2}}+T_{4\kappa_{1}})|\phi^{\prime}|^{p-2}\phi^{\prime}=-\bar{\mu}_{1}(m,p,\kappa_{1},\kappa_{2},D)|\phi|^{p-2}\phi

on $[0,D/2]$ with $\phi(s)>0$ for $s\in(0,D/2]$ , $\phi^{\prime}(s)>0$ for $s\in[0,D/2)$ , and $\phi^{\prime}(D/2)=0$ .

Proof.

From the proof of Lemma 3.1, we see that there exists an odd function $\phi$ corresponding to $\bar{\mu}_{1}(m,p,\kappa_{1},\kappa_{2},D)$ with $\phi(0)=0$ and $\phi^{\prime}(D/2)=0$ , which does not change sign in $(0,D/2]$ . Since equation (3.16) is equivalent to

(|\phi^{\prime}|^{p-2}\phi^{\prime}c_{\kappa_{2}}^{2m-2}c_{4\kappa_{1}})^{\prime}=-\bar{\mu}_{1}c_{\kappa_{2}}^{2m-2}c_{4\kappa_{1}}|\phi|^{p-2}\phi

and $\phi^{\prime}(D/2)=0$ , then $\phi^{\prime}(s)>0$ on $[0,D/2)$ . $\square$

Now we turn to prove Theorem 1.2.

Proof of Theorem 1.2.

For any $D_{1}>D$ , let $\bar{\mu}_{1}=\bar{\mu}_{1}(m,p,\kappa_{1},\kappa_{2},D_{1})$ be the first nonzero Neumann eigenvalue of the eigenvalue problem (1.5). By Lemma 3.2, there exists an odd eigenfunction $\phi(s)$ corresponding to $\bar{\mu}_{1}$ such that $\phi(s)>0$ on $(0,D_{1}/2]$ , $\phi^{\prime}(s)>0$ on $[0,D_{1}/2)$ and $\phi^{\prime}(D_{1}/2)=0$ . Let $u(x)$ be an eigenfucntion corresponding to $\mu_{1,p}(M)$ , and it is well known that $u(x)\in C^{1,\alpha}(M)$ for some $\alpha\in(0,1)$ , then there exists $C_{1}>0$ such that

(3.17)

|u(y)-u(x)|\leq C_{1}d(x,y)

for $x,y\in M$ . Noting that $\varphi^{\prime}$ is positive on $[0,D/2]$ , then from equation (3.16) we deduce that $\varphi$ is smooth on $[0,D/2]$ , and for all $x,y\in M$ it holds

(3.18)

\phi(\frac{d(x,y)}{2})\geq C_{2}d(x,y)

for some $C_{2}>0$ . By (3.17) and (3.18), we can choose a $C>0$ such that

|u(y)-u(x)|\leq 2C\phi(\frac{d(x,y)}{2})

for all $x,y\in M$ .

Let

\displaystyle v(x,t)=\exp(-\mu_{1,p}^{\frac{1}{p-1}}t)u(x),

and

\displaystyle\varphi(s,t)=C\exp(-\bar{\mu}_{1}^{\frac{1}{p-1}}t)\phi(s).

It is easy to verify that $v$ and $\varphi$ satisfy the conditions in Theorem 3.1, and then it holds

v(y,t)-v(x,t)\leq 2\varphi(\frac{d(x,y)}{2},t)

for all $t>0$ , namely

\exp(-\mu_{1,p}^{\frac{1}{p-1}}t)\big{(}u(y)-u(x)\big{)}\leq 2C\exp(-\bar{\mu}_{1}^{\frac{1}{p-1}}t)\phi(\frac{d(x,y)}{2}).

As $t\rightarrow\infty$ , the above inequality gives

\mu_{1,p}\geq\bar{\mu}_{1}(m,p,\kappa_{1},\kappa_{2},D_{1}),

then Theorem 1.2 follows by letting $D\rightarrow D_{1}$ . $\square$

4. The First Dirichlet Eigenvalue

In this section, we give the proof of Theorem 1.4. The key tools are a comparison theorem for distance to the boundary and Barta’s inequality. Let $M$ be a compact manifold with smooth boundary $\partial M$ , and define the distance function to the boundary of $M$ by

d(x,\partial M)=\inf\Big{\{}d(x,y):y\in\partial M\Big{\}}.

By choosing $\alpha=(p-1)|\nabla\psi|^{p-2}$ and $\beta=|\nabla\psi|^{p-2}$ in Theorem 6.1 of [14], we have the following lemma.

Lemma 4.1.

Let $(M^{m},g,J)$ be a compact Kähler manifold with smooth boundary $\partial M$ . Suppose that $H\geq 4\kappa_{1}$ and $\operatorname{Ric}^{\perp}\geq 2(m-1)\kappa_{2}$ for some $\kappa_{1},\kappa_{2}\in\mathbb{R}$ , and the second fundamental form on $\partial M$ is bounded from below by $\Lambda\in\mathbb{R}$ . Let $p\in(1,\infty)$ , and assume $\varphi$ is a smooth function on $[0,R]$ satisfying $\varphi^{\prime}\geq 0$ . Then for any smooth function $\psi$ satisfying

\psi(x)\leq\varphi(d(x,\partial M))\quad\text{for}\quad x\in M,\quad\text{and}\quad\psi(x_{0})=\varphi(d(x_{0},\partial M)),

it holds

\Delta_{p}\psi(x_{0})\leq\bar{\mathfrak{L}}\varphi(d(x_{0},\partial M)),

where one-dimensional operator $\bar{\mathfrak{L}}$ is defined by

(4.1)

\displaystyle\bar{\mathfrak{L}}\varphi=(p-1)|\varphi^{\prime}|^{p-2}\varphi^{\prime\prime}-\big{(}2(m-1)T_{\kappa_{2},\Lambda}+T_{4\kappa_{1},\Lambda}\big{)}|\varphi^{\prime}|^{p-2}\varphi^{\prime}

for all $\varphi\in C^{2}([0,R])$ , and $T_{\kappa,\Lambda}$ is defined by (1.8).

Consider the following one-dimensional eigenvalue problem on $[0,R]$

(4.2)

\left\{\begin{aligned} &\bar{\mathfrak{L}}\varphi=-\lambda|\varphi|^{p-2}\varphi,\\ &\varphi(0)=0,\ \varphi^{\prime}(R)=0,\end{aligned}\right.

and denote by $\bar{\lambda}_{1}(m,p,\kappa_{1},\kappa_{2},D)$ , written as $\bar{\lambda}_{1}$ for short, the first eigenvalue of problem (4.2). Then it is easy to see that $\bar{\lambda}_{1}$ can be characterized by

\bar{\lambda}_{1}=\inf\bigg{\{}\frac{\int_{0}^{R}|\phi^{\prime}|^{p}C_{\kappa_{2},\Lambda}^{2m-2}C_{4\kappa_{1},\Lambda}\ \mathrm{d}s}{\int_{0}^{R}|\phi|^{p}C_{\kappa_{2},\Lambda}^{2m-2}C_{4\kappa_{1},\Lambda}\ \mathrm{d}s}\ \big{|}\phi\in W^{1,p}\big{(}(0,R)\big{)}\backslash\{0\},\ \phi(0)=0\bigg{\}},

where $C_{\kappa,\Lambda}$ is defined by (1.7).

Proof of Theorem 1.4.

Let $\varphi$ be an eigenfunction corresponding to $\bar{\lambda}_{1}$ , then

\bar{\mathfrak{L}}\varphi=-\bar{\lambda}_{1}|\varphi|^{p-2}\varphi

with $\varphi(0)=0$ and $\varphi^{\prime}(R)=0$ . Similarly as in the proof of Lemma 3.2, we can choose $\varphi$ such that $\varphi(s)>0$ on $(0,R]$ , and $\varphi^{\prime}(s)>0$ on $[0,R)$ . Define a trial function for $\lambda_{1,p}(M)$ by

v(x):=\varphi(d(x,\partial M)),

then Lemma 4.1 gives

(4.3)

\displaystyle\Delta_{p}v(x)\leq\bar{\mathfrak{L}}\varphi(d(x,\partial M))

away from the cut locus of $\partial M$ , and thus globally in the distributional sense, see [13, Lemma 5.2]. Recall that $\varphi$ is an eigenfunction with respect to $\bar{\lambda}_{1}$ , namely

(4.4)

\displaystyle\bar{\mathfrak{L}}\varphi(d(x,\partial M))=-\bar{\lambda}_{1}|v(x)|^{p-2}v(x),

thus we conclude from (4.3) and (4.4) that

\displaystyle\Delta_{p}v\leq-\bar{\lambda}_{1}|v|^{p-2}v,\quad x\in M.

By the definition of $v(x)$ , we see $v(x)>0$ for $x\in M$ , and $v(x)=0$ for $x\in\partial M$ . Then using Barta’s inequality (cf. [13, Theorem 3.1]), we have

\lambda_{1,p}(M)\geq\bar{\lambda}_{1},

proving Theorem 1.4. $\square$

Lastly, we emphasize that when $\kappa_{1}=\kappa_{2}=\kappa$ , a standard argument (see for instance [13, Section 6]) indicates that when $M$ is a $(\kappa,\Lambda)$ -model space in the complex space forms, the first Dirichlet eigenfunction of the $p$ -Laplacian can be written in the form $u=\varphi\circ d(x,\partial M)$ , and $\varphi$ is the first eigenfunction of one-dimensional problem (4.2), which gives the equality case of Theorem 1.2. Therefore estimate (1.9) is sharp when $\kappa_{1}=\kappa_{2}$ .

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Lower bounds for the first eigenvalue of pp-Laplacian on Kähler manifolds

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction and Main Results

Theorem 1.1 ([14]).

Theorem 1.2.

Corollary 1.3.

Theorem 1.4.

2. Preliminary

2.1. Curvatures of Kähler Manifolds

Remark 2.1.

2.2. Modulus of Continuity

3. The First Nonzero Eigenvalue

Theorem 3.1.

Proof.

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Proof of Theorem 1.2.

4. The First Dirichlet Eigenvalue

Lemma 4.1.

Proof of Theorem 1.4.

References

Lower bounds for the first eigenvalue of $p$ -Laplacian on Kähler manifolds