Results 1  10
of
176
On the relation between elliptic and parabolic Harnack inequalities
, 2001
"... We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in que ..."
Abstract

Cited by 66 (6 self)
 Add to MetaCart
We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1
Harnack inequalities and subGaussian estimates for random walks
 Math. Annalen
, 2002
"... We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the m ..."
Abstract

Cited by 60 (5 self)
 Add to MetaCart
We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the mean exit time in any ball of radius R is of the order R . The latter condition can be replaced by a certain estimate of a resistance of annuli.
The DirichlettoNeumann map for complete Riemannian manifolds with boundary
 Comm. Geom. Anal
"... Abstract. We study the problem of determining a complete Riemannian manifold with boundary from the Cauchy data of harmonic functions. This problem arises in electrical impedance tomography, where one tries to ¯nd an unknown conductivity inside a given body from measurements done on the boundary of ..."
Abstract

Cited by 53 (29 self)
 Add to MetaCart
Abstract. We study the problem of determining a complete Riemannian manifold with boundary from the Cauchy data of harmonic functions. This problem arises in electrical impedance tomography, where one tries to ¯nd an unknown conductivity inside a given body from measurements done on the boundary of the body. Here, we show that one can reconstruct a complete, realanalytic, Riemannian manifold M with compact boundary from the set of Cauchy data, given on a nonempty open subset ¡ of the boundary, of all harmonic functions with Dirichlet data supported in ¡, provided dim M ¸ 3. We note that for this result we need no assumption on the topology of the manifold other than connectedness, nor do we need a priori knowledge of all
REMARKS ON NONCOMPACT GRADIENT RICCI SOLITONS
, 905
"... Abstract. In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and L pLiouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality ..."
Abstract

Cited by 43 (8 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and L pLiouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scalar curvature estimates for gradient Ricci solitons under L p conditions on the relevant quantities.
Heat Kernel and Essential Spectrum of Infinite Graphs
"... ABSTRACT. We study the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs. For general graphs, a uniqueness criterion, shown to be optimal, is given in terms of the maximal valence on spheres about a fixed vertex. A sufficient condition for nonuniqueness is al ..."
Abstract

Cited by 31 (1 self)
 Add to MetaCart
(Show Context)
ABSTRACT. We study the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs. For general graphs, a uniqueness criterion, shown to be optimal, is given in terms of the maximal valence on spheres about a fixed vertex. A sufficient condition for nonuniqueness is also presented. Furthermore, we give a lower bound on the bottom of the spectrum of the discrete Laplacian and use this bound to give a condition ensuring that the essential spectrum of the Laplacian is empty. 1.
SaloffCoste L., Stability results for Harnack inequalities
"... We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain nonuniform changes of the weight. We also prove necessary and sufficient conditions for the Har ..."
Abstract

Cited by 27 (2 self)
 Add to MetaCart
We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain nonuniform changes of the weight. We also prove necessary and sufficient conditions for the Harnack inequalities to hold on complete noncompact manifolds having nonnegative Ricci curvature outside a compact set and a finite first Betti number or just having asymptotically nonnegative sectional curvature. Contents
Complete proper minimal surfaces in convex bodies of R³ (II): The behavior of the limit set
, 2005
"... ..."
(Show Context)
Stable constant mean curvature surfaces
 Advanced Lectures in Mathematics
, 2008
"... We study relationships between stability of constant mean curvature surfaces in a Riemannian threemanifold N and the geometry of leaves of laminations and foliations of N by surfaces of possibly varying constant mean curvature (the case of minimal leaves is included as well). Many of these results ..."
Abstract

Cited by 23 (1 self)
 Add to MetaCart
(Show Context)
We study relationships between stability of constant mean curvature surfaces in a Riemannian threemanifold N and the geometry of leaves of laminations and foliations of N by surfaces of possibly varying constant mean curvature (the case of minimal leaves is included as well). Many of these results extend to the case of codimension one laminations and foliations in ndimensional Riemannian manifolds by hypersurfaces of possibly varying constant mean curvature. Since this contribution is for a handbook in Differential Geometry, we also describe some of the basic theory of CMC (constant mean curvature) laminations and some of the new techniques and results which we feel will have an impact on the subject in future years.