On Riemannian Manifolds With Positive Weighted Ricci Curvature of Negative Effective Dimension

Abstract

In this paper, we investigate complete Riemannian manifolds satisfying the lower weighted Ricci curvature bound Ric N ≥ K with K > 0 for the negative effective dimension N < 0. We analyze two one-dimensional examples of constant curvature Ric N ≡ K with finite and infinite total volumes. We also discuss when the first non-zero eigenvalue of the Laplacian takes its minimum under the same condition Ric N ≥ K > 0, as a counterpart to the classical Obata rigidity theorem. Our main theorem shows that, if N < −1 and the minimum is attained, then the manifold splits off the real line as a warped product of hyperbolic nature.

Stability estimates for the sharp spectral gap bound under a curvature-dimension condition

“…Since the Bochner inequality in the non-smooth setting seems not to have been investigated yet in the literature (in particular, its possible equivalence with other non-smooth definitions studied in [41]), we shall restrict ourselves to the smooth setting, and assume the Bochner inequality (1.1) holds in a strong sense, as in [32]. As noted in [31], the spectral gap bound of Theorem 2.2 is sharp for N ⩽ −1, but ceases to be sharp when N is negative and |N | is small (see also [34] where spaces with negative effective dimension, positive curvature and infinite volume are studied). When N ⩽ −1, a particular example of a model satisfying a CD(1 − N, N ) condition is given by the generalized Cauchy distribution as presented in Section 2.3,…”

“…Sharp functional inequalities and model spaces in this setting were studied for example in [38]. Rigidity in the smooth setting was studied in [34]. Under this curvature condition, the manifold may have infinite volume, so we shall require finite volume as an extra condition.…”