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Register a new userSub-Riemannian Ricci curvature via generalized Gamma $z$ calculus
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We derive sub-Riemannian Ricci curvature tensor for sub-Riemannian manifolds. We provide examples including the Heisenberg group, displacement group ($textbf{SE}(2)$), and Martinet sub-Riemannian structure with arbitrary weighted volumes, in which we establish analytical bounds for sub-Riemannian curvature dimension bounds and log-Sobolev inequalities. {These bounds can be used to establish the entropy dissipation results for sub-Riemannian drift diffusion processes on a compact spatial domain, in term of $L_1$ distance.} Our derivation of Ricci curvature is based on generalized Gamma $z$ calculus and $z$--Bochners formula, where $z$ stands for extra directions introduced into the sub-Riemannian degenerate structure.
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We generalize the Gamma $z$ calculus to study degenerate drift-diffusion processes, where $z$ stands for extra directions introduced into the degenerate system. Based on this calculus, we establish the sub-Riemannian Ricci curvature tensor and the associated curvature dimension bound for general sub-Riemannian manifolds. These results do not require the commutative iteration of Gamma and Gamma z operator and go beyond the step two condition. These allow us to analyze the convergence properties of degenerate drift-diffusion processes and prove the entropy dissipation rate and several functional inequalities in sub-Riemannian manifolds. Several examples are provided. In particular, we show the global in time convergence result for displacement group with a weighted volume on a compact region. The new Gamma $z$ calculus is motivated by optimal transport and density manifold. We embed the probability density space over sub-Riemannian manifold with the $L^2$ sub-Riemannian Wasserstein metric. We call it sub-Riemannian density manifold (SDM). We study the dynamical behavior of the degenerate Fokker-Planck equation as gradient flows in SDM. Our derivation builds an equivalence relation between Gamma z calculus and second-order calculus in SDM.
We examine questions of geometric realizability for algebraic structures which arise naturally in affine and Riemannian geometry. Suppose given an algebraic curvature operator R at a point P of a manifold M and suppose given a real analytic (resp. C-k for finite k at least 2) pseudo-Riemannian metric on M defined near P. We construct a torsion free real analytic (resp. C-k) connection D which is defined near P on the tangent bundle of M whose curvature operator is the given operator R at P and so that D has constant scalar curvature. We show that if R is Ricci symmetric, then D can be chosen to be Ricci symmetric; if R has trace free Ricci tensor, then D can be chosen to have trace free Ricci tensor; if R is Ricci alternating, then D can be chosen to be Ricci alternating.
We give a proof of the fact that the upper and the lower sectional curvature bounds of a complete manifold vary at a bounded rate under the Ricci flow.
We revisit classical eigenvalue inequalities due to Buser, Cheng, and Gromov on closed Riemannian manifolds, and prove t
In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in $sf RCD$ spaces where the reference measure is the Hausdorff measure. A quasi-perimeter is a functional given by the sum of the usual perimeter and of a suitable continuous term. In particular, isoperimetric sets are a particular case of our study. We prove that on an ${sf RCD}(K,N)$ space $({rm X},{sf d},mathcal{H}^N)$, with $Kinmathbb R$, $Ngeq 2$, and a uniform bound from below on the volume of unit balls, volume constrained minimizers of quasi-perimeters are open bounded sets with $(N-1)$-Ahlfors regular topological boundary coinciding with the essential boundary. The proof is based on a new Deformation Lemma for sets of finite perimeter in ${sf RCD}(K,N)$ spaces $({rm X},{sf d},mathfrak m)$ and on the study of interior and exterior points of volume constrained minimizers of quasi-perimeters. The theory applies to volume constrained minimizers in smooth Riemannian manifolds, possibly with boundary, providing a general regularity result for such minimizers in the smooth setting.
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