169 research outputs found
Foliation by free boundary constant mean curvature leaves
Let be a Riemannian manifold of dimension with smooth boundary and
. We prove that there exists a smooth foliation around
whose leaves are submanifolds of dimension , constant mean curvature and its
arrive perpendicular to the boundary of M, provided that is a nondegenerate
critical point of the mean curvature function of
Eigenvalue estimates for submanifolds with locally bounded mean curvature
We give lower bounds for the first Dirichilet eigenvalues for domains in
submanifolds with locally bounded mean curvatures. These bounds depend on the
injectivity radius, sectional curvature (upperbound) of the ambient space and
on the mean curvature of the submanifold. For submanifolds fo Hadamard
manifolds these lower bounds depend only on the dimension and mean curvature of
the submanifold.Comment: Paper writen in latex, 7 page
On the mean curvature of Nash isometric embeddings
J. Nash proved that the geometry of any Riemannian manifold M imposes no
restrictions to be embedded isometrically into a (fixed) ball
B_{\mathbb{R}^{N}}(1) of the Euclidean space R^N. However, the geometry of M
appears, to some extent, imposing restrictions on the mean curvature vector of
the embedding.Comment: A note of two page
An Extension of Barta's Theorem and Geometric Applications
We prove an extension of a theorem of Barta then we make few geometric
applications. We extend Cheng's lower eigenvalue estimates of normal geodesic
balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds
of the space forms. We prove an stability theorem for minimal hypersurfaces of
the Euclidean space, giving a converse statement of a result of Schoen. Finally
we prove a generalization of a result of Kazdan-Kramer about existence of
solutions of certain quasi-linear elliptic equations.Comment: 23 pages. This paper is an improved version of our paper of the same
Titled posted her
On compact CMC-Hypersurfaces of
Let be the set of all closed
-hypersurfaces , where is a simply
connected complete Riemannian -manifold with sectional curvature . We show that {\Hm}(N\times \mathbb{R})=\inf_{M\in {\mathscr
F}(N\times \mathbb{R})}\{| H_{M}| \}\geq (n-1)\kappa/n .Comment: We changed the title from a Remark on compact CMC-Hypersurfaces of
to On compact and we added another theorem to the paper
(theorem 1.4
On Cheng's Eigenvalue Comparison Theorems
We prove Cheng's eigenvalue comparison theorems for geodesic balls within the
cut locus under weaker geometric hypothesis, and we also show that there are
certain geometric rigidity in case of equality of the eigenvalues. This
rigidity becomes isometric rigidity under upper sectional curvature bounds or
lower Ricci curvature bounds. We construct examples of smooth metrics showing
that our results are true extensions of Cheng's theorem. We also construct a
family of complete smooth metrics on the Euclidean space non-isometric to the
constant sectional curvature k metrics of the simply connected space forms of
constant sectional curvature k such that the geodesic balls of radius r have
the same first eigenvalue and the geodesic spheres have the same mean
curvatures. In the end we construct examples of Riemannian manifolds M with
arbitrary topology with positive fundamental tone positive that generalize
Veeravalli's examples.Comment: paper with 11 page
Essential spectrum of a class of Riemannian manifolds
In this paper we consider a family of Riemannian manifolds, not necessarily
complete, with curvature conditions in a neighborhood of a ray. Under these
conditions we obtain that the essential spectrum of the Laplacian contains an
interval. The results presented in this paper allow to determine the spectrum
of the Laplace operator on unlimited regions of space forms, such as horoball
in hyperbolic space and cones in Euclidean space
Riemannian submersions with discrete spectrum
We prove some estimates on the spectrum of the Laplacian of the total space
of a Riemannian submersion in terms of the spectrum of the Laplacian of the
base and the geometry of the fibers. When the fibers of the submersions are
compact and minimal, we prove that the total space is discrete if and only if
the base is discrete. When the fibers are not minimal, we prove a discreteness
criterion for the total space in terms of the relative growth of the mean
curvature of the fibers and the mean curvature of the geodesic spheres in the
base. We discuss in particular the case of warped products
An estimate for the sectional curvature of cylindrically bounded submanifolds
We give sharp sectional curvature estimates for complete immersed
cylindrically bounded -submanifolds ,
provided that either is proper with the second
fundamental form with certain controlled growth or has scalar curvature
with strong quadratic decay. This latter gives a non-trivial extension of the
Jorge-Koutrofiotis Theorem [7]Comment: To appear in the Transactions of the American Mathematical Societ
Complete subamanifolds of with finite topology
We show that a complete -dimensional immersed submanifold of
with is properly immersed and have finite topology,
where is an scaling invariant number that gives the rate
that the norm of the second fundamental form decays to zero at infinity. The
class of submanifolds with contains all complete minimal surfaces
in with finite total curvature, all -dimensional minimal
submanifolds of with finite total scalar curvature
and all complete 2-dimensional
complete surfaces with and nonpositive
curvature with respect to every normal direction, since for them.Comment: 8 page
- β¦