Willmore submanifolds in a sphere

Let $x:M\to S^{n+p}$ be an $n$-dimensional submanifold in an $(n+p)$-dimensional unit sphere $S^{n+p}$, $x:M\to S^{n+p}$ is called a Willmore submanifold to the following Willmore functional: $$\int_M(S-nH^2)^{\frac{n}{2}}dv,$$ where $S=\sum\limits_{\alpha,i,j}(h^\alpha_{ij})^2$ is the square of the length of the second fundamental form, $H$ is the mean curvature of $M$. In [13], author proved an integral inequality of Simon's type for $n$-dimensional compact Willmore hypersurfaces in $S^{n+1}$ and gave a characterization of {\it Willmore tori}. In this paper, we generalize this result to $n$-dimensional compact Willmore submanifolds in $S^{n+p}$. In fact, we obtain an integral inequality of Simon's type for compact Willmore submanifolds in $S^{n+p}$ and give a characterization of {\it willmore tori} and {\it Veronese surface} by use of integral inequality.Comment: 18 pages. To appear in Mathematical Research Lette

Second Eigenvalue of Paneitz Operators and Mean Curvature

For $n\geq 7$, we give the optimal estimate for the second eigenvalue of Paneitz operators for compact $n$-dimensional submanifolds in an $(n+p)$-dimensional space form

The sharp estimates for the first eigenvalue of Paneitz operator on 4-dimensional submanifolds

In this note, we obtain the sharp estimates for the first eigenvalue of Paneitz operator for $4$-dimensional compact submanifolds in Euclidean space. Since unit spheres and projective spaces can be canonically imbedded into Euclidean space, the corresponding estimates for the first eigenvalue are also obtained

On inverse mean curvature flow in Schwarzschild space and Kottler space

In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface $\Sigma$ is strictly mean convex and star-shaped, then the flow hypersurface $\Sigma_t$ converges to a large coordinate sphere as $t\rightarrow \infty$ exponentially. We also describe an application of this convergence result. In the second part of this paper, we will analyse the inverse mean curvature flow in Kottler-Schwarzchild manifold. By deriving a lower bound for the mean curvature on the flow hypersurface independently of the initial mean curvature, we can use an approximation argument to show the global existence and regularity of the smooth inverse mean curvature flow for star-shaped and weakly mean convex initial hypersurface, which generalizes Huisken-Ilmanen's result [18].Comment: 23 pages, v2, title changed, new result adde

Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian

We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: $$u_t=\Delta_\phi(u^p)$$ associated with the Witten Laplacian on Riemannian manifolds. Under the assumption that the $m$-dimensional Bakry-Emery Ricci curvature is bounded from below, we obtain gradient estimates which generalize the results in [20] and [13]. Moreover, inspired by X. -D. Li's work in [19] we also study the entropy formulae introduced in [20] for porous medium equations and fast diffusion equations associated with the Witten Laplacian. We prove monotonicity theorems for such entropy formulae on compact Riemannian manifolds with non-negative $m$-dimensional Bakry-Emery Ricci curvatureComment: 25 page

Calabi product Lagrangian immersions in complex projective space and complex hyperbolic space

Starting from two Lagrangian immersions and a Legendre curve $\tilde{\gamma}(t)$ in $\mathbb{S}^3(1)$ (or in $\mathbb{H}_1^3(1)$), it is possible to construct a new Lagrangian immersion in $\mathbb{CP}^n$ (or in $\mathbb{CH}^n$), which is called a warped product Lagrangian immersion. When $\tilde{\gamma}(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, r_2e^{i(- \frac{r_1}{r_2}at)})$ (or $\tilde{\gamma}(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, r_2e^{i(\frac{r_1}{r_2}at)})$), where $r_1$, $r_2$, and $a$ are positive constants with $r_1^2+r_2^2=1$ (or $-r_1^2+r_2^2=-1$), we call the new Lagrangian immersion a Calabi product Lagrangian immersion. In this paper, we study the inverse problem: how to determine from the properties of the second fundamental form whether a given Lagrangian immersion of $\mathbb{CP}^n$ or $\mathbb{CH}^n$ is a Calabi product Lagrangian immersion. When the Calabi product is minimal, or is Hamiltonian minimal, or has parallel second fundamental form, we give some further characterizations

New characterizations of the Clifford torus as a Lagrangian self-shrinker

In this paper, we obtain several new characterizations of the Clifford torus as a Lagrangian self-shrinker. We first show that the Clifford torus $\mathbb{S}^1(1)\times\mathbb{S}^1(1)$ is the unique compact orientable Lagrangian self-shrinker in $\mathbb{C}^2$ with $|A|^2\leq 2$, which gives an affirmative answer to Castro-Lerma's conjecture. We also prove that the Clifford torus is the unique compact orientable embedded Lagrangian self-shrinker with nonnegative or nonpositive Gauss curvature in $\mathbb{C}^2$.Comment: 16 pages, accepted by The Journal of Geometric Analysi

Second eigenvalue of a Jacobi operator of hypersurfaces with constant scalar curvature

Let $x:M\to\mathbb{S}^{n+1}(1)$ be an n-dimensional compact hypersurface with constant scalar curvature $n(n-1)r,~r\geq 1$, in a unit sphere $\mathbb{S}^{n+1}(1),~n\geq 5$. We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral $\int_MH dv$ of the mean curvature $H$. In this paper, we derive an optimal upper bound for the second eigenvalue of the Jacobi operator $J_s$ of $M$. Moreover, when $r>1$, the bound is attained if and only if $M$ is totally umbilical and non-totally geodesic, when $r=1$, the bound is attained if $M$ is the Riemannian product $\mathbb{S}^{m}(c)\times\mathbb{S}^{n-m}(\sqrt{1-c^2}),~1\leq m\leq n-2,~c=\sqrt{\frac{(n-1)m+\sqrt{(n-1)m(n-m)}}{n(n-1)}}$.Comment: Corrected typo

Stability of capillary hypersurfaces in a Euclidean ball

We study the stability of capillary hypersurfaces in a unit Euclidean ball. It is proved that if the mass center of the generalized body enclosed by the immersed capillary hypersurface and the wetted part of the sphere is located at the origin, then the hypersurface is unstable. An immediate result is that all known examples except the totally geodesic ones and spherical caps are unstable.Comment: 15 pages, 1 figure; all comments are welcom

The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature

In 1963, K.P.Grotemeyer proved an interesting variant of the Gauss-Bonnet Theorem. Let M be an oriented closed surface in the Euclidean space R^3 with Euler characteristic \chi(M), Gauss curvature G and unit normal vector field n. Grotemeyer's identity replaces the Gauss-Bonnet integrand G by the normal moment ^2G, where $a$ is a fixed unit vector. Grotemeyer showed that the total integral of this integrand is (2/3)pi times chi(M). We generalize Grotemeyer's result to oriented closed even-dimesional hypersurfaces of dimension n in an (n+1) ndimensional space form N^{n+1}(k).Comment: 10 page