Heat kernel on smooth metric measure spaces with nonnegative curvature

Abstract

Abstract. We derive a local Gaussian upper bound for the f -heat kernel on complete smooth metric measure space (M, g, e −f dv) with nonnegative Bakry-Émery Ricci curvature. As applications, we obtain a sharp L 1 f -Liouville theorem for f -subharmonic functions and an L 1 f -uniqueness property for nonnegative solutions of the f -heat equation, assuming f is of at most quadratic growth. In particular, any L 1 f -integrable f -subharmonic function on gradient shrinking and steady Ricci solitons must be constant. We also provide explicit f -heat kernel for Gaussian solitons.

Heat kernel on smooth metric measure spaces and applications

“…By a similar argument in [23] (see also [48]), we also prove an L 1 f -uniqueness theorem for solutions of f -heat equation, see Theorem 5.3 in Section 5.…”

“…Indeed, let (R, g 0 , e −f dx) be a 1-dimensional steady Gaussian soliton, where g 0 is the Euclidean metric and f (x) = ±x. From [48] the f -heat kernel is given by…”

Liouville-type theorems on the complete gradient shrinking Ricci solitons

“…For any complete smooth metric measure space (M, g, e −f dv), Pigola, Rimoldi and Setti proved the L p Liouville theorem for p > 1, see [15]. For the case of p = 1, Wu [19] proved the L 1 Liouville theorem under the assumptions f is bounded and Ric f ≥ −C(1 + d 2 (x)); without any assumption, is proved by Wu-Wu (see Corollary 1.6 in [20]). The Corollary 0.11 was proved by Pigola-Rimoldi-Setti, see Theorem 25 in [16].…”

“…If we remove the condition, the result is false. One easy example is provided by Wu-Wu (see Example 1.8 in [20]), as following Remark 0.10. For any complete smooth metric measure space (M, g, e −f dv), Pigola, Rimoldi and Setti proved the L p Liouville theorem for p > 1, see [15].…”

Elliptic gradient estimates for a nonlinear heat equation and applications

“…If u is independent of time t, then it is f -harmonic function. In the past few years, various Liouville properties for f -harmonic functions were obtained, see for example [3], [16], [17], [21], [23], [25], [27], [28], and the references therein. Recently, the author [26] proved elliptic gradient estimates and parabolic Liouville properties for f -heat equation under some assumptions of (∞-)Bakry-Émery Ricci tensor.…”