1,720,980 research outputs found
Rigidity and gap results for low index properly immersed self-shrinkers in Rm+1
Full text linkIn this paper we show that the only properly immersed self-shrinkers Σin Rm+1with Morse index 1are the hyperplanes through the origin. Moreover, we prove that if Σis not a hyperplane through the origin then the index jumps and it is at least m +2, with equality if and only if Σis a cylinder Rm−k×Sk(√k)for some 1 ≤k≤m −
Stability properties and topology at infinity of f-minimal hypersurfaces
Full text linke study stability properties of f–minimal hypersurfaces isometrically immersed in weighted manifolds with non–negative Bakry–Emery Ricci ́
curvature under volume growth conditions. Moreover, exploiting a weighted
version of a finiteness result and the adaptation to this setting of Li–Tam
theory, we investigate the topology at infinity of f–minimal hypersurfaces.
On the way, we prove a new comparison result in weighted geometry and we
provide a general weighted L
1–Sobolev inequality for hypersurfaces in Cartan–
Hadamard weighted manifolds, satisfying suitable restrictions on the weight
function
On the growth of supersolutions of nonlinear PDE's on exterior domains
No full textWe obtain a comparison principle on annuli, with “catenoid-like” functions, for supersolutions of non-linear elliptic PDEs over exterior domains in a non-positively curved manifold with a pole. This result is applied to get an upper estimate on the growth of such supersolutions and, in particular, of exterior graphs of non-negative mean curvatur
Poincaré Inequality and Topological Rigidity of Translators and Self-Expanders for the Mean Curvature Flow
Full text linkWe prove an abstract structure theorem for weighted manifolds supporting a weighted
f -Poincaré inequality and whose ends satisfy a suitable non-integrability condition.
We then study how our arguments can be used to obtain full topological control on two
important classes of hypersurfaces of the Euclidean space, namely translators and selfexpanders for the mean curvature flow, under either stability or curvature asumptions.
As an important intermediate step in order to get our results we get the validity of a
Poincaré inequality with respect to the natural weighted measure on any translator and
we prove that any end of a translator must have infinite weighted volume. Similar tools
can be obtained for properly immersed self-expanders permitting to get topological
rigidity under curvature assumptions
Height estimates and half-space theorems for spacelike hypersurfaces in generalized Robertson-Walker spacetimes
No full textAsymptotically non-negative Ricci curvature, elliptic Kato constant and isoperimetric inequalities
No full textThe ABP method for proving isoperimetric inequalities has been first employed by Cabré
in R n, then developed by Brendle, notably in the context of non-compact Riemannian
manifolds of non-negative Ricci curvature and positive asymptotic volume ratio. In this
paper, we expand upon their approach and prove isoperimetric inequalities (sharp in the
limit) in the presence of a small amount of negative curvature. First, we consider smallness of the negative part Ric− of the Ricci curvature in terms of its elliptic Kato constant.
Indeed, the Kato constant turns out to control the non-negativity of the (∞)- Bakry-Émery
Ricci-tensor of a suitable conformal deformation of the manifold, and the ABP method can
be implemented in this setting. Secondly, we show that the smallness of the Kato constant
is ensured provided that the asymptotic volume ratio is positive and either M has one end
and asymptotically non-negative sectional curvature, or there is a suitable polynomial
decay of Ric−, and the relative volume comparison condition known as (VC) holds. To
show this latter fact, we enhance techniques elaborated by Li-Tam and Kasue to obtain
new estimates of the Green function valid on the whole manifol
The Frankel property for self-shrinkers from the viewpoint of elliptic PDEs
Full text linkWe show that two properly embedded self-shrinkers in Euclidean space that are sufficiently separated at infinity must intersect at a finite point. The proof is based on a localized version of the Reilly formula applied to a suitable f-harmonic function with controlled gradient. In the immersed case, a new direct proof of the generalized half-space property is also presente
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