We show that, if a certain Sobolev inequality holds, then a scale-invariant 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

Geometrically based methods for various tasks of machine learning have attracted considerable attention over the last few years. In this paper we show convergence of eigenvectors of the point cloud Laplacian to the eigenfunctions of the Laplace-Beltrami operator on the underlying manifold, thus establishing the first convergence results for a spectral dimensionality reduction algorithm in the manifold setting. 1

Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and

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Abstract. In [26], Perelman established a differential Li-Yau-Hamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds (also see [23]). As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete non-compact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. We also prove a long time existence result for the Kähler-Ricci flow flow on complete non-negatively curved Kähler manifolds.

this paper is to estimate the hitting probability function / K (t; x) := P x (9 s 2 [0; t] ; X s 2 K) where K ae M is a fixed compact set. In words, / K (t; x) is the probability that Brownian motion started at x hits K by time t. Our goal is to obtain precise estimates on / K for all t ? 0 and x outside a neighborhood of K, hence avoiding the somewhat different question of the behavior of / K near the boundary of K. In the context of Riemannian manifolds, this natural question has been considered only in a handful of papers including [2], [4]. We were led to study / K in our attempt to develop sharp heat kernel estimates on manifolds with more than one end. Indeed, the proof of the heat kernel estimates announced in [20] depends in a crucial way on the results of the present paper (see [21]). In this context, it turns out to be important to estimate also the time derivative @ t / K (t; x) which is a positive function. We develop a general approach which allows to obtain estimates of / K in terms of the heat kernel p(t; x; y) or closely related objects such as the Dirichlet heat kernel p U (t; x; y) of some open set U . In the case when X t is transient, that is, M is non-parabolic, we show that the behavior of / K (t; x), away from K, is comparable to that of Z t 0 p(s; x; y)ds; where y is a reference point on @K. If (X t ) t?0 is recurrent, that is, M is parabolic, we obtain similar estimates through Z t 0 p U (s; x; y)ds where U is a certain region slightly larger than\Omega := M n K. We also show that @ t / K (t; x) is comparable to p\Omega (t; x; y) where y stays at a certain distance from @K. For precise statements, see Theorems 3.3, 3.5, 3.7 and Corollaries 3.9, 3.10. Using the known results concerning the heat kernel p(t; x; y) and the results of [23...

Abstract. A new type of gradient estimate is established for diffusion semigroups on non-compact complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived for diffusion semigroups on arbitrary complete Riemannian manifolds. 1. The main result Let M be a non-compact complete Riemannian manifold, and Pt be the Dirichlet diffusion semigroup generated by L = ∆ + ∇V for some C 2 function V. We intend to establish reasonable gradient estimates and Harnack type inequalities for Pt. In case that Ric − HessV is bounded below, a dimension-free Harnack inequality was established in [15], which according to [17], is indeed equivalent to the corresponding curvature condition. See e.g. [2] for equivalent statements on heat kernel functional inequalities; see also [8, 3, 9] for a parabolic Harnack inequality using the dimensioncurvature condition by shifting time, which goes back to the classical local parabolic Harnack inequality of Moser [10]. Recently, some sharp gradient estimates have been derived in [13, 19] for the Dirichlet semigroup on relatively compact domains. More precisely, for V = 0 and a relatively compact open C2 domain D, the Dirichlet heat semigroup P D t satisfies (1.1) |∇P D t f|(x) ≤ C(x, t)P D t f(x), x ∈ D, t> 0, for some locally bounded function C: D ×]0, ∞ [ →]0, ∞ [ and all f ∈ B + b, the space of bounded non-negative measurable functions on M. Obviously, this implies the Harnack inequality (1.2) P D t f(x) ≤ ˜ C(x, y, t)P D t f(y), t> 0, x, y ∈ D, f ∈ B+ b, for some function ˜ C: M 2 ×]0, ∞ [ →]0, ∞[. The purpose of this paper is to establish inequalities analogous to (1.1) and (1.2) globally on the whole manifold M. On the other hand however, both (1.1) and (1.2) are in general wrong for Pt in place of P D t. A simple counter-example is already the standard heat semigroup on R d. Hence, we turn to search for the following slightly weaker version of gradient

On considère la classe des variétés riemanniennes complètes non compactes dont le noyau de la chaleur satisfait une estimation supérieure et inférieure gaussienne. On montre que la transformée de Riesz y est bornée sur L p, pour un intervalle ouvert de p au-dessus de 2, si et seulement si le gradient du noyau de la chaleur satisfait une certaine estimation L p pour le même intervalle d’exposants p. One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L p bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain L p estimate in the same

Abstract. In this paper, we prove a general maximum principle for the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we construct complete manifolds with bounded nonnegative sectional curvature of dimension greater than or equal to four such that the Ricci flow does not preserve the nonnegativity of the sectional curvature, even though the nonnegativity of the sectional curvature was proved to be preserved by Hamilton in dimension three. The example is the first of this type. This fact is proved through a general splitting theorem on the complete family of metrics with nonnegative sectional curvature, deformed by the Ricci flow. §0 Introduction. The Ricci flow has been proved to be an effective tool in the study of the geometry and topology of manifolds. One of the good properties of the Ricci flow is that it preserves the ‘nonnegativity ’ of the curvature. In dimension three, Hamilton [ H1] proves that on compact manifolds the Ricc flow preserves the nonnegativity of the Ricci curvature and

The purpose of this work is to describe an abstract theory of Hardy-Sobolev spaces on doubling Riemannian manifolds via an atomic decomposition. We study the real interpolation of these spaces with Sobolev spaces and finally give applications to Riesz transforms.