240 research outputs found
Willmore submanifolds in a sphere
Let be an -dimensional submanifold in an
-dimensional unit sphere , is called a
Willmore submanifold to the following Willmore functional: where
is the square of the length of
the second fundamental form, is the mean curvature of . In [13], author
proved an integral inequality of Simon's type for -dimensional compact
Willmore hypersurfaces in and gave a characterization of {\it
Willmore tori}. In this paper, we generalize this result to -dimensional
compact Willmore submanifolds in . In fact, we obtain an integral
inequality of Simon's type for compact Willmore submanifolds in 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 , we give the optimal estimate for the second eigenvalue of
Paneitz operators for compact -dimensional submanifolds in an
-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 -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 is
strictly mean convex and star-shaped, then the flow hypersurface
converges to a large coordinate sphere as 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: associated
with the Witten Laplacian on Riemannian manifolds. Under the assumption that
the -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
-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
in (or in ), it is
possible to construct a new Lagrangian immersion in (or in
), which is called a warped product Lagrangian immersion. When
(or ), where , , and are positive
constants with (or ), 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 or
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
is the unique compact orientable
Lagrangian self-shrinker in with , 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
.Comment: 16 pages, accepted by The Journal of Geometric Analysi
Second eigenvalue of a Jacobi operator of hypersurfaces with constant scalar curvature
Let be an n-dimensional compact hypersurface with
constant scalar curvature , in a unit sphere
. We know that such hypersurfaces can be
characterized as critical points for a variational problem of the integral
of the mean curvature . In this paper, we derive an optimal
upper bound for the second eigenvalue of the Jacobi operator of .
Moreover, when , the bound is attained if and only if is totally
umbilical and non-totally geodesic, when , the bound is attained if is
the Riemannian product
.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 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
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