Upper bound of Kähler angles on the β-symplectic critical surfaces

Abstract

Let (M,g) be a Kähler surface and \mathrm{\Sigma } be a \beta-symplectic critical surface in M. If L_q\left(\mathrm{\Sigma }\right) is bounded for some q>3, then we give a uniform upper bound for the Kähler angle on \mathrm{\Sigma }. This bound only depends on M,q,\beta and the L_q functional of \mathrm{\Sigma }. For q>4, this estimate is known and we extend the scope of q.

Keywords

Kähler surface / β-symplectic critical surfaces / Kähler angle / L_β functional

1 Introduction

The holomorphic curve in Kähler surface is very important in Differential Geometry. To investigate the existence and properties of it, many methods and concepts are used. The \beta-symplectic critical surface is one of them. Roughly speaking, \beta-symplectic critical surfaces can be considered as the generation of the holomorphic curves and minimal surfaces. For more works on the \beta-symplectic critical surfaces, one can see ([2-4,8,10,11]). In form, one can expect to obtain a holomorphic curve from a sequence of \beta-symplectic critical surfaces when \beta \to \mathrm{\infty }. The convergence is much complicate. For the blow-up analysis during the convergence, there are many beautiful works. But it is open that whether the genus remain invariance during the convergence. This is one of the most important problems in this field. To investigate the procedure, some geometric quantities need uniform estimates during the convergence. In this note, our main purpose is to general the uniform upper bound for the Kähler angle during the convergence \beta \to \mathrm{\infty }.

Suppose that M is a compact Kähler surface with Kähler form \omega and a compatible complex structure J. The Riemannian metric g on M is defined by

g(U,V) =\omega (U,JV).

Let \mathrm{\Sigma } be a real compact oriented surface without boundary that is smoothly immersed in a Kähler surface M. In [2] Chern and Wolfson defined the Kähler angle \alpha of \mathrm{\Sigma } in M as

\omega {|}_{\mathrm{\Sigma }}=\cos \alpha \mathrm{d}{V}_{\mathrm{\Sigma }}.

Here \mathrm{d}{V}_{\mathrm{\Sigma }} is the area element of \mathrm{\Sigma } with induced metric g from M. It is easy to see that \alpha is continuous on \mathrm{\Sigma } and it is smooth away from the points \alpha =0,\pi.

If \cos \alpha \equiv 1, then \mathrm{\Sigma } is a holomorphic curve of M. On the other hand, if \cos \alpha \equiv 0, then \mathrm{\Sigma } is a minimal surface of M. \mathrm{\Sigma } is called a symplectic surface if \cos \alpha >0 everywhere on \mathrm{\Sigma }. For a symplectic surface \mathrm{\Sigma } and \beta >0, the L_{\beta } functional is defined as

L_{\beta } (\mathrm{\Sigma } )={\int }_{\mathrm{\Sigma }}\frac{1}{{\cos }^{\beta }\alpha }\mathrm{d}{V}_{\mathrm{\Sigma }}.

In [3], Han and Li firstly gave the Euler−Lagrange equation of L_1 functional

{\cos }^3\alpha \mathit{H}=( J(J\mathrm{\nabla }\cos \alpha {)}^{\mathrm{\top }}{)}^{\mathrm{\perp }}.

\mathrm{\Sigma } is called a symplectic critical surface if it satisfies the above Euler−Lagrange equation. They obtained some results by variation and flow (one can see the Webster formula in [9]).

To investigate the existence of the holomorphic curves in a Kähler surface, in [5], Han, Li and Sun considered the L_{\beta } functional and proved the following theorem.

Theorem 1.1 [5, Theorem 2.1]　 Suppose that M is a compact Kähler surface and \mathrm{\Sigma } is a real surface in M. For any smooth vector field X on \mathrm{\Sigma }, the first variation formula of the functional L_{\beta } is

{\delta }_X L_{\beta }=-(\beta +1){\int }_{\mathrm{\Sigma }}\frac{X\cdot \mathit{H}}{{\cos }^{\beta }\alpha }\mathrm{d}{V}_{\mathrm{\Sigma }}+\beta (\beta +1){\int }_{\mathrm{\Sigma }}\frac{X\cdot (J(J\mathrm{\nabla }\cos \alpha {)}^{\mathrm{\top }}{)}^{\mathrm{\perp }}}{{\cos }^{\beta +3}\alpha }\mathrm{d}{V}_{\mathrm{\Sigma }},

where \mathit{H} is the mean curvature vector of \mathrm{\Sigma } in the Kähler surface M, ({)}^{\mathrm{\top }} means tangential components of () and ({)}^{\mathrm{\perp }} means the normal components of (). The Euler-Lagrange equation of L_{\beta } is

(1.1){\cos }^3\alpha \mathit{H}-\beta (J(J \mathrm{\nabla }\cos \alpha {)}^{\mathrm{\top }}{)}^{\mathrm{\perp }}= 0.

A surface satisfying equation (1.1) is called \beta-symplectic critical surface.

If , then (\cos \alpha {)}^{-\beta }\to \mathrm{\infty } as \beta \to \mathrm{\infty }. Therefore, in form, when \beta \to \mathrm{\infty }, if the \beta-symplectic critical surfaces {\mathrm{\Sigma }}_{\beta }\to \mathrm{\Sigma } in some sense, then one can expect that \alpha =0 on \mathrm{\Sigma } and thus \mathrm{\Sigma } is a holomorphic curve. On the other hand, it is easy to see that L_{\beta } \to {\int }_{\mathrm{\Sigma }}\mathrm{d}{V}_{\mathrm{\Sigma }}=S_{\mathrm{\Sigma }} when \beta \to 0. So, one can expect that a sequence \beta-symplectic critical surfaces convergence to a minimal surface in some sense when \beta \to 0.

In [5], Han, Li and Sun proved that if the scalar curvature of \mathrm{\Sigma } is positive and \mathrm{\Sigma } is a \beta-symplectic critical surface for all \beta \ge 0, then it is a minimal surface with constant Kähler angle. But, if the scalar curvature is negative, this result is not true (see [1]).

Soon afterwards, in [6], Han, Li and Sun obtained more results in the existence of the holomorphic curves in a Kähler surface. They proved that for a sequence of \beta-symplectic critical surfaces, when \beta \to \mathrm{\infty }, there exists a subsequence which converge to a holomorphic curve weakly. But it is unknown that whether the genus (Euler characteristic) remain invariance during the convergence. To study this problem, they gave a uniform upper bound on the Kähler angle \alpha (i.e., a uniform lower bound on \cos \alpha) for \beta-symplectic critical surfaces.

Theorem 1.2 [6, Theorem 3.1]　 Assume that \mathrm{\Sigma } is a close \beta-symplectic critical surface in a compact Kähler surface M. If for some q>4, then there exists a positive constant \delta which only depends on \beta ,q,M,L_q(\mathrm{\Sigma }) such that

\underset{\mathrm{\Sigma }}{inf}\cos \alpha \ge \delta >0.

Motivated by these works, we revise some computations and improve Theorem 1.2. In detail, we replace the condition q>4 in Theorem 1.2 by q>3. The following theorem is the main result in this note.

Theorem 1.3　 Use the same notations in the above theorem. If for some q>3, then there exists a positive constant \delta which only depends on \beta ,q,M, L_q(\mathrm{\Sigma }) such that

(1.2)\underset{\mathrm{\Sigma }}{inf}\cos \alpha \ge \delta >0.

Remark　In the proof, we get an iteration L^{p+1}\to L^{2p-1}. As f\in L^q, only if q>3, we can take p_0=q-1> 2 such that 2p_0-1>p_0+1 and obtain a bound on \Vert f{\Vert }_{L^{\mathrm{\infty }}} by iterations. It is easy to see that our proof cannot run if q\le 3.

Throughout this note we use C to denote a positive constant which only depends on \beta ,q,\mathrm{\Sigma },M,L^q(\mathrm{\Sigma }) and it may vary from line to line.

2 Some fundamental lemmas

Let M be a smooth Kähler surface without boundary, \mathrm{\Sigma } be a smooth oriented surface in an M without boundary, \mathit{H} be the mean curvature vector of \mathrm{\Sigma } in M and \alpha be the Kähler angle of \mathrm{\Sigma } in M.

At first we need the Sobolev inequality in a manifold.

Lemma 2.1 [7, Theorem 2.1]　 Suppose that \mathrm{\Sigma } is an n-dimensional submanifold in {\mathbb{R}}^K and \mathit{H} is the mean curvature vector of \mathrm{\Sigma } in {\mathbb{R}}^K. Then for any u\in C^1(\mathrm{\Sigma }), there holds

(2.1)\Vert u{\Vert }_{L^{\frac{n}{n-1}}(\mathrm{\Sigma })}\le C_n(\Vert \mathrm{\nabla }u{\Vert }_{L^1(\mathrm{\Sigma })}+\Vert u\mathit{H}{\Vert }_{L^1(\mathrm{\Sigma })}),

where C_n only depends on n and is independent of \mathrm{\Sigma }.

Remark　If \mathrm{\Sigma } is a compact submanifold in M, then by Nash embedding theorem, M can be isometric embedded into {\mathbb{R}}^K when K is big enough. We use \mathit{H} and \overline{\mathit{H}} to denote the mean curvature vector of \mathrm{\Sigma } in M and {\mathbb{R}}^K. By some simple computations we can get that

\overline{\mathit{H}}=\mathit{H}+\mathit{E},

where \mathit{E} only depends on the second fundamental form of M in {\mathbb{R}}^K and |\mathit{E}| can be controlled by a constant C_M that only depends on M. Thus, when \mathrm{\Sigma } and M are compact, we can use the mean curvature vector \mathit{H} of \mathrm{\Sigma } in M to replace the mean curvature vector \overline{\mathit{H}} of \mathrm{\Sigma } in {\mathbb{R}}^K.

In this note, we also need the following form of Sobolev inequality.

Lemma 2.2　 Use the same notations and assumptions in Lemma 2.1. For any a\ge 1 we have

(2.2)\Vert u{\Vert }_{L^{\frac{na}{n-1}}(\mathrm{\Sigma })}^a\le C_n(a\Vert u{\Vert }_{L^{2(a-1)}(\mathrm{\Sigma })}^{a-1}\Vert \mathrm{\nabla }u {\Vert }_{L^2(\mathrm{\Sigma })}+\Vert |u{|}^a\mathit{H}{\Vert }_{L^1(\mathrm{\Sigma })}).

Proof　In (2.1), we use | u{|}^a to replace u. From Hölder inequality we obtain that

\begin{array}{rl}\Vert u{\Vert }_{L^{\frac{na}{n-1}}(\mathrm{\Sigma })}^a& =\Vert |u{|}^a{\Vert }_{L^{\frac{n}{n-1}}(\mathrm{\Sigma })}\\ & \le C_n(\Vert \mathrm{\nabla }|u{|}^a{\Vert }_{L^1(\mathrm{\Sigma })}+\Vert |u{|}^a\mathit{H}{\Vert }_{L^1(\mathrm{\Sigma })})\\ & \le C_n(a\Vert |u{|}^{a-1}\mathrm{\nabla }u{\Vert }_{L^1(\mathrm{\Sigma })}+\Vert |u{|}^a\mathit{H}{\Vert }_{L^1(\mathrm{\Sigma })})\\ & \le C_n(a\Vert |u{|}^{a-1}{\Vert }_{L^2(\mathrm{\Sigma })}\Vert \mathrm{\nabla }u{\Vert }_{L^2(\mathrm{\Sigma })}+\Vert |u{|}^a\mathit{H}{\Vert }_{L^1(\mathrm{\Sigma })})\\ & =C_n(a\Vert u{\Vert }_{L^{2(a-1)}(\mathrm{\Sigma })}^{a-1}\Vert \mathrm{\nabla }u{\Vert }_{L^2(\mathrm{\Sigma })}+\Vert |u{|}^a\mathit{H}{\Vert }_{L^1(\mathrm{\Sigma })}).\end{array}

Thus the proof is completed.□

Besides, we also need the following two important inequalities for the \beta-symplectic critical surfaces.

Lemma 2.3　 Suppose that f=\frac{1}{\cos \alpha } and | K_M|\le K_0(>1), where K_M is the section curvature of M. Then we have

(2.3)|\mathit{H}|\le \beta |\mathrm{\nabla } f|;

(2.4)\mathrm{\Delta }f \ge -K_0{f}^2.

Proof　These inequalities can be found in [6] and here we give some simple illustrations.

At first, by using the result in [6, Section 2] and the Euler-Lagrange equation (1.1), one gets that

|H|=\beta \frac{{\sin }^2\alpha }{{\cos }^2\alpha }|\mathrm{\nabla }\alpha | .

Here V={\mathrm{\nabla }}_{e_2}\alpha e_3+{\mathrm{\nabla }}_{e_1}\alpha e_4 in the orthogonal coordinate system.

Then, as

\mathrm{\nabla }f =-\frac{\mathrm{\nabla }\cos \alpha }{{\cos }^2\alpha }=\frac{\sin \alpha }{{\cos }^2\alpha }\mathrm{\nabla }\alpha ,

we have

|H|=\beta \sin \alpha \frac{\sin \alpha }{{\cos }^2\alpha }|\mathrm{\nabla } \alpha |=\beta \sin \alpha |\mathrm{\nabla }f|\le \beta |\mathrm{\nabla } f|.

Therefore we obtain the first inequality.

Secondly, there holds [6, Theorem 2.3]

\begin{array}{rl}\mathrm{\Delta }\cos \alpha =& \frac{2\beta {\sin }^2\alpha }{\cos \alpha ({\cos }^2\alpha +\beta {\sin }^2\alpha )}|\mathrm{\nabla }\alpha {|}^2-2\cos \alpha |\mathrm{\nabla }\alpha {|}^2\\ & -\frac{{\cos }^2\alpha {\sin }^2\alpha }{{\cos }^2\alpha +\beta {\sin }^2\alpha }Ric(J{e}_1,e_2).\end{array}

By some direct computations one has

\begin{array}{rl}\mathrm{\Delta }f& =\mathrm{\Delta }\frac{1}{\cos \alpha }\\ & =\frac{2{\sin }^2\alpha }{{\cos }^3\alpha }|\mathrm{\nabla }\alpha {|}^2-\frac{\mathrm{\Delta }\cos \alpha }{{\cos }^2\alpha }\\ & =\left(\frac{2{\sin }^2\alpha }{{\cos }^3\alpha }-\frac{2\beta {\sin }^2\alpha }{{\cos }^3\alpha ({\cos }^2\alpha +\beta {\sin }^2\alpha )}+\frac{2}{\cos \alpha }\right)|\mathrm{\nabla }\alpha {|}^2\\ & +\frac{{\sin }^2\alpha }{{\cos }^2\alpha +\beta {\sin }^2\alpha }Ric(J{e}_1,e_2)\\ & =\frac{2}{\cos \alpha }|\mathrm{\nabla }\alpha {|}^2+\frac{{\sin }^2\alpha }{{\cos }^2\alpha +\beta {\sin }^2\alpha }Ric(J{e}_1,e_2)\\ & \ge -\frac{{\sin }^2\alpha }{{\cos }^2\alpha +\beta {\sin }^2\alpha }{K}_0\\ & =-\frac{{\cos }^2\alpha {\sin }^2\alpha }{{\cos }^2\alpha +\beta {\sin }^2\alpha }{K}_0{f}^2\\ & \ge -K_0{f}^2.\end{array}

Thus we complete the proof of this lemma.□

3 Proof of the main theorem

At first, for any x_0\in \mathrm{\Sigma }, it is no matter to consider (2.4) in a geodesic ball D_2(x_0) in \mathrm{\Sigma }. Let \eta \in C_0^{\mathrm{\infty }}(D_2(x_0)) be a nonnegative cut-off function which will be determined below.

For any p>2, multiply {\eta }^2 f^{p-1} on the both sides of (2.4). From the integration by parts, we can get that

\begin{array}{rl}-K_0{\int }_{\mathrm{\Sigma }}{\eta }^2{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}& \le {\int }_{\mathrm{\Sigma }}{\eta }^2{f}^{p-1}\mathrm{\Delta }f\mathrm{d}{V}_{\mathrm{\Sigma }}\\ & =-{\int }_{\mathrm{\Sigma }}\mathrm{\nabla }({\eta }^2{f}^{p-1})\mathrm{\nabla }f\mathrm{d}{V}_{\mathrm{\Sigma }}\\ & =-2{\int }_{\mathrm{\Sigma }}\eta f^{p-1}\mathrm{\nabla }\eta \mathrm{\nabla }f\mathrm{d}{V}_{\mathrm{\Sigma }}-(p-1){\int }_{\mathrm{\Sigma }}{\eta }^2{f}^{p-2}|\mathrm{\nabla }f{|}^2\mathrm{d}{V}_{\mathrm{\Sigma }}.\end{array}

Besides, since f=\frac{1}{\cos \alpha }\ge 1, from the above inequality one can obtain that

\begin{array}{rl}& (p-1){\int }_{\mathrm{\Sigma }}{\eta }^2{f}^{p-2}|\mathrm{\nabla }f{|}^2\mathrm{d}{V}_{\mathrm{\Sigma }}\\ & \le -2{\int }_{\mathrm{\Sigma }}\eta f^{p-1}\mathrm{\nabla }\eta \mathrm{\nabla }f\mathrm{d}{V}_{\mathrm{\Sigma }}+K_0{\int }_{\mathrm{\Sigma }}{\eta }^2{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\\ & \le \frac{p-1}{2}{\int }_{\mathrm{\Sigma }}{\eta }^2{f}^{p-2}|\mathrm{\nabla }f{|}^2\mathrm{d}{V}_{\mathrm{\Sigma }}+\frac{2}{p-1}{\int }_{\mathrm{\Sigma }}|\mathrm{\nabla }\eta {|}^2{f}^p\mathrm{d}{V}_{\mathrm{\Sigma }}+K_0{\int }_{\mathrm{\Sigma }}{\eta }^2{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\\ & \le \frac{p-1}{2}{\int }_{\mathrm{\Sigma }}{\eta }^2{f}^{p-2}|\mathrm{\nabla }f{|}^2\mathrm{d}{V}_{\mathrm{\Sigma }}+{\int }_{\mathrm{\Sigma }}\left(\frac{2|\mathrm{\nabla }\eta {|}^2}{p-1}+K_0{\eta }^2\right)f^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}.\end{array}

Eliminating the first term, we yield that

(3.1)(p-1){\int }_{\mathrm{\Sigma }}{\eta }^2{f}^{p-2}|\mathrm{\nabla }f {|}^2\mathrm{d}{V}_{\mathrm{\Sigma }}\le {\int }_{\mathrm{\Sigma }}\left(\frac{4|\mathrm{\nabla }\eta {|}^2}{p-1}+2K_0{\eta }^2\right) f^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}.

By using the fact p>2,f\ge 1, (3.1), and some simple computations, we obtain that

(3.2)\begin{array}{rl}{\int }_{\mathrm{\Sigma }}|\mathrm{\nabla }\left(\eta f^{\frac{p}{2}}\right){|}^2\mathrm{d}{V}_{\mathrm{\Sigma }}& \le 2{\int }_{\mathrm{\Sigma }}(|\mathrm{\nabla }\eta {|}^2{f}^p+\frac{p^2}{4}{\eta }^2{f}^{p-2}|\mathrm{\nabla }f{|}^2)\mathrm{d}{V}_{\mathrm{\Sigma }}\\ & \le 2{\int }_{\mathrm{\Sigma }}|\mathrm{\nabla }\eta {|}^2{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}+\frac{p^2}{p-1}{\int }_{\mathrm{\Sigma }}\left(\frac{2|\mathrm{\nabla }\eta {|}^2}{p-1}+K_0\eta \right)f^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\\ & \le {\int }_{\mathrm{\Sigma }}\left(2|\mathrm{\nabla }\eta {|}^2+\frac{2p^2|\mathrm{\nabla }\eta {|}^2}{(p-1{)}^2}+\frac{K_0{p}^2\eta }{p-1}\right)f^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\\ & \le {\int }_{\mathrm{\Sigma }}\left(\frac{2p^2|\mathrm{\nabla }\eta {|}^2}{(p-1{)}^2}+\frac{2p^2|\mathrm{\nabla }\eta {|}^2}{(p-1{)}^2}+\frac{K_0{p}^2\eta }{p-1}\right)f^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\\ & ={\int }_{\mathrm{\Sigma }}\left(\frac{4|\mathrm{\nabla }\eta {|}^2}{p-1}+K_0\eta \right)\frac{p^2}{p-1}{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}.\end{array}

When n=2,p>2, a=2-\frac{1}{p}, taking \eta f^{\frac{p}{2}}=u in Lemma 2.2, one yields that

(3.3)\Vert \eta f^{\frac{p}{2}} {\Vert }_{L^{4-\frac{2}{p}}(\mathrm{\Sigma })}^{2-\frac{1}{p}}\le C\left(\left(2-\frac{1}{p}\right)\Vert \eta f^{\frac{p}{2}}{\Vert }_{L^{2-\frac{2}{p}}(\mathrm{\Sigma })}^{1-\frac{1}{p}}\Vert \mathrm{\nabla }(\eta f^{\frac{p}{2}}){\Vert }_{L^2(\mathrm{\Sigma })}+\Vert (\eta f^{\frac{p}{2}}{)}^{2-\frac{1}{p}}\mathit{H}{\Vert }_{L^1(\mathrm{\Sigma })}\right).

For the last term of (3.3), from (2.3) we have

(3.4)\begin{array}{rl}& \Vert (\eta f^{\frac{p}{2}}{)}^{2-\frac{1}{p}}\mathit{H}{\Vert }_{L^1(\mathrm{\Sigma })}\\ & \le \beta \Vert (\eta f^{\frac{p}{2}}{)}^{2-\frac{1}{p}}\mathrm{\nabla }f{\Vert }_{L^1(\mathrm{\Sigma })}\\ & =\frac{2\beta }{p}\Vert {\eta }^{1-\frac{1}{p}}{f}^{\frac{p}{2}+\frac{1}{2}}\eta \mathrm{\nabla }(f^{\frac{p}{2}}){\Vert }_{L^1(\mathrm{\Sigma })}\\ & =\frac{2\beta }{p}\Vert {\eta }^{1-\frac{1}{p}}{f}^{\frac{p}{2}+\frac{1}{2}}(\mathrm{\nabla }(\eta f^{\frac{p}{2}})-f^{\frac{p}{2}}\mathrm{\nabla }\eta ){\Vert }_{L^1(\mathrm{\Sigma })}\\ & \le \beta \Vert {\eta }^{1-\frac{1}{p}}{f}^{\frac{p}{2}+\frac{1}{2}}{\Vert }_{L^2(\mathrm{\Sigma })}(\Vert \mathrm{\nabla }(\eta f^{\frac{p}{2}}){\Vert }_{L^2(\mathrm{\Sigma })}+\Vert f^{\frac{p}{2}}\mathrm{\nabla }\eta {\Vert }_{L^2(\mathrm{\Sigma })})\\ & =\beta {\left({\int }_{\mathrm{\Sigma }}{\eta }^{2-\frac{2}{p}}{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\right)}^{\frac{1}{2}}\left(\Vert \mathrm{\nabla }(\eta f^{\frac{p}{2}}){\Vert }_{L^2(\mathrm{\Sigma })}+{\left({\int }_{\mathrm{\Sigma }}|\mathrm{\nabla }\eta {|}^2{f}^p\mathrm{d}{V}_{\mathrm{\Sigma }}\right)}^{\frac{1}{2}}\right)\\ & \le \beta {\left({\int }_{\mathrm{\Sigma }}\eta f^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\right)}^{\frac{1}{2}}\left(\Vert \mathrm{\nabla }(\eta f^{\frac{p}{2}}){\Vert }_{L^2(\mathrm{\Sigma })}+{\left({\int }_{\mathrm{\Sigma }}|\mathrm{\nabla }\eta {|}^2{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\right)}^{\frac{1}{2}}\right),\end{array}

where we use the fact p>2 and f\ge 1 in the second and last inequalities.

Now from (3.4) and (3.2), we yield that

(3.5)\begin{array}{rl}& \Vert (\eta f^{\frac{p}{2}}{)}^{2-\frac{1}{p}}\mathit{H}{\Vert }_{L^1(\mathrm{\Sigma })}\\ & \le \beta {\left({\int }_{\mathrm{\Sigma }}\eta f^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\right)}^{\frac{1}{2}}\left(\Vert \mathrm{\nabla }(\eta f^{\frac{p}{2}}){\Vert }_{L^2(\mathrm{\Sigma })}+{\left({\int }_{\mathrm{\Sigma }}|\mathrm{\nabla }\eta {|}^2{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\right)}^{\frac{1}{2}}\right)\\ & \le \beta {\left({\int }_{\mathrm{\Sigma }}\eta f^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\right)}^{\frac{1}{2}}({\left({\int }_{\mathrm{\Sigma }}\left(\frac{4|\mathrm{\nabla }\eta {|}^2}{p-1}+K_0\eta \right)\frac{p^2}{p-1}{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\right)}^{\frac{1}{2}}\\ & +{\left({\int }_{\mathrm{\Sigma }}|\mathrm{\nabla }\eta {|}^2{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\right)}^{\frac{1}{2}})\\ & \le 2\beta {\left({\int }_{\mathrm{\Sigma }}\eta f^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\right)}^{\frac{1}{2}}{\left({\int }_{\mathrm{\Sigma }}\left(\frac{4|\mathrm{\nabla }\eta {|}^2}{p-1}+K_0\eta \right)\frac{p^2}{p-1}{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\right)}^{\frac{1}{2}}\\ & \le 2\beta {\int }_{\mathrm{\Sigma }}\left(\frac{4|\mathrm{\nabla }\eta {|}^2}{p-1}+K_0\eta \right)\frac{p^2}{p-1}{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}.\end{array}

In the last inequality we use the assumption \frac{K_0{p}^2}{p-1}>1.

By using (3.3), (3.2) and (3.5), we have

(3.6)\begin{array}{rl}\Vert \eta f^{\frac{p}{2}}{\Vert }_{L^{4-\frac{2}{p}}(\mathrm{\Sigma })}^{2-\frac{1}{p}}& \le C(\Vert \eta f^{\frac{p}{2}}{\Vert }_{L^{2-\frac{2}{p}}(\mathrm{\Sigma })}^{1-\frac{1}{p}}\Vert \mathrm{\nabla }(\eta f^{\frac{p}{2}}){\Vert }_{L^2(\mathrm{\Sigma })}+\Vert (\eta f^{\frac{p}{2}}{)}^{2-\frac{1}{p}}\mathit{H}{\Vert }_{L^1(\mathrm{\Sigma })})\\ & \le C(({\int }_{\mathrm{\Sigma }}{\eta }^{2-\frac{2}{p}}{f}^{p-1}\mathrm{d}{V}_{\mathrm{\Sigma }}{)}^{\frac{1}{2}}({\int }_{\mathrm{\Sigma }}(\frac{4|\mathrm{\nabla }\eta {|}^2}{p-1}+K_0\eta )\frac{p^2}{p-1}{f}^{p+1}d{V}_{\mathrm{\Sigma }}{)}^{\frac{1}{2}}\\ & +\Vert (\eta f^{\frac{p}{2}}{)}^{2-\frac{1}{p}}\mathit{H}{\Vert }_{L^1(\mathrm{\Sigma })})\\ & \le C(({\int }_{\mathrm{\Sigma }}\eta f^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}{)}^{\frac{1}{2}}({\int }_{\mathrm{\Sigma }}(\frac{4|\mathrm{\nabla }\eta {|}^2}{p-1}+K_0\eta )\frac{p^2}{p-1}{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}{)}^{\frac{1}{2}}\\ & +2\beta {\int }_{\mathrm{\Sigma }}(\frac{4|\mathrm{\nabla }\eta {|}^2}{p-1}+K_0\eta )\frac{p^2}{p-1}{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }})\\ & \le C\beta {\int }_{\mathrm{\Sigma }}(\frac{4|\mathrm{\nabla }\eta {|}^2}{p-1}+K_0\eta )\frac{p^2}{p-1}{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}.\end{array}

Now, we define D_k=D_{1+2^{1-k}}( x_0) and choose a cut-off function {\eta }_k\in C_0^{\mathrm{\infty }}(D_k) on \mathrm{\Sigma } such that

0\le {\eta }_k\le 1; {\eta }_k =1\text{on}{D}_{k+1};|\mathrm{\nabla }{\eta }_k|\le C{2}^k.

Set p_k=2^{k-1}(q-3 )+3 and p=p_k-1. Then 2p-1=p_{k+1}. From (3.6), we obtain that

(3.7)\begin{array}{rl}{\left({\int }_{D_{k+1}}{f}^{p_{k+1}}\mathrm{d}{V}_{\mathrm{\Sigma }}\right)}^{\frac{1}{2}}& \le {\left({\int }_{\mathrm{\Sigma }}{\eta }_k^{4-\frac{2}{p}}{f}^{2p-1}\mathrm{d}{V}_{\mathrm{\Sigma }}\right)}^{\frac{1}{2}}\\ & =\Vert {\eta }_k{f}^{\frac{p}{2}}{\Vert }_{L^{4-\frac{2}{p}}(\mathrm{\Sigma })}^{2-\frac{1}{p}}\\ & \le C_1\beta {\int }_{\mathrm{\Sigma }}\left(\frac{4|\mathrm{\nabla }{\eta }_k{|}^2}{p-1}+K_0{\eta }_k\right)\frac{p^2}{p-1}{f}^{p+1}\mathrm{d}{V}_{\mathrm{\Sigma }}\\ & =C_1\beta {\int }_{\mathrm{\Sigma }}\left(\frac{4|\mathrm{\nabla }{\eta }_k{|}^2}{2^{k-1}(q-3)+1}+K_0{\eta }_k\right)\frac{(2^{k-1}(q-3)+2{)}^2}{2^{k-1}(q-3)+1}{f}^{p_k}\mathrm{d}{V}_{\mathrm{\Sigma }}\\ & \le C_{\beta }{\int }_{D_k}(2^k+K_0)2^k{f}^{p_k}\mathrm{d}{V}_{\mathrm{\Sigma }}\\ & \le C_2{4}^k{\int }_{D_k}{f}^{p_k}\mathrm{d}{V}_{\mathrm{\Sigma }},\end{array}

where C_2=C_2(M,\beta ,q) only depends on M,q,\beta.

Set t_k=\ln \Vert f{\Vert }_{L^{p_k}(D_k)}. From (3.7) one yields that

\begin{array}{rl}{t}_{k+1}& =\ln {\left({\int }_{D_{k+1}}{f}^{p_{k+1}}\mathrm{d}{V}_{\mathrm{\Sigma }}\right)}^{\frac{1}{p_{k+1}}}\\ & =\frac{2}{p_{k+1}}\ln {\left({\int }_{D_{k+1}}{f}^{p_{k+1}}\mathrm{d}{V}_{\mathrm{\Sigma }}\right)}^{\frac{1}{2}}\\ & \le \frac{2}{p_{k+1}}(\ln C_2+k\ln 4+p_k{t}_k)\\ & =\frac{2\ln C_2+k\ln 16}{2^k(q-3)+3}+\left(1+\frac{3}{2^k(q-3)+3}\right)t_k\\ & \le C_{3}k{2}^{-k}+(1+C_4{2}^{-k})t_k\\ & \le C_5{2}^{-\frac{k}{2}}+(1+C_5{2}^{-\frac{k}{2}})t_k.\end{array}

So we have

(3.8)\begin{array}{rl}{t}_{k+1}+1& \le (1+C_5{2}^{-\frac{k}{2}})(t_k+1)\\ & \le \prod _{i=1}^k(1+C_5{2}^{-\frac{i}{2}})(t_1+1)\\ & \le C_6\ln (\mathrm{e}\Vert f{\Vert }_{L^q(D_2(x_0))}),\end{array}

where C_5,C_6 only depend on M,q,\beta.

Therefore, we can yield that

\begin{array}{rl}\Vert f{\Vert }_{L^{\mathrm{\infty }}(D_1(x_0))}& =\underset{k\to \mathrm{\infty }}{lim}\Vert f{\Vert }_{L^{p_{k+1}}(D_{k+1})}\\ & ={\mathrm{e}}^{\underset{k\to \mathrm{\infty }}{lim}{t}_{k+1}}\\ & \le {\mathrm{e}}^{C_6\ln (\mathrm{e}\Vert f{\Vert }_{L^q(D_2(x_0))})}\\ & ={\mathrm{e}}^{C_6}\Vert f{\Vert }_{L^q(D_2(x_0))}^{C_6}.\end{array}

As f=\frac{1}{\cos \alpha }, we get that

\Vert \frac{1}{\cos \alpha }{\Vert }_{L^{\mathrm{\infty }}(D_1(x_0))}\le {\mathrm{e}}^{C_6}\Vert \frac{1}{\cos \alpha }{\Vert }_{L^q(D_2(x_0))}^{C_6}.

Take \delta ={\mathrm{e}}^{-C_6}{L}_q^{-\frac{C_6}{q}}(\mathrm{\Sigma }), we obtain that

\begin{array}{l}\underset{\mathrm{\Sigma }}{inf}\cos \alpha \ge \delta >0.\end{array}

Therefore, we complete the proof.□

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