arXiv research

Researchers develop weighted GJMS operators for smooth metric measure spaces.

The aim of this note is to study the measure-valued Ricci tensor on smooth metric measure space with boundary, which is a generalization of Bakry-Emery's modified Ricci tensor on weighted Riemannian manifold. As an application, we offer a new approach to study curvature-dimension condition of smooth metric measure spac…

Paper finds lower bounds for eigenvalues of Bi-drifted Laplacian on smooth metric measure spaces.

Locally homogeneous RCD spaces are shown to be smooth manifolds.

Solves Ricci flow on Riemann surfaces with measure initial data.

Proves existence and uniqueness of weighted metrics for smooth spaces.

The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily …

Study shows non-spectrality of certain curves and line segments.

The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.

For smooth metric measure spaces $(M, g, e^{-f} dvol)$ we prove a Liuoville-type theorem when the Bakry-Emery Ricci tensor is nonnegative. This generalizes a result of Yau, which is recovered in the case $f$ is constant. This result follows from a gradient estimate for f-harmonic functions on smooth metric measure spac…

A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem

Estimates smooth graph signals from partial measurements.

Unified positive mass theorem and Dirac operator study on weighted manifolds.

We introduce and study the notion of the energy of a smooth metric measure space, which includes as special cases the Yamabe constant and Perelman's $ν$-entropy. We then investigate some properties the energy shares with these constants, in particular its relationship with the $κ$-noncollapsing property. Finally, we us…

In this paper, we prove almost Schur Lemma on closed smooth metric measure spaces, which implies the results of X. Cheng and De Lellis-Topping whenever the weighted function f is constant.

The paper provides gradient estimates for nonlinear heat-type equations on smooth metric measure spaces.

We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positiv…

The paper provides new gradient estimates for solutions to a nonlinear elliptic equation on smooth metric measure spaces.

Let $\{T^t\}$ be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let $μ$ be an ergodic measure of maximal entropy. We show that either $\{T^t\}$ is Bernoulli, or $\{T^t\}$ is isomorphic to the product of a Bernoulli flow and a rotational flow. Appli…

For non-smooth surfaces, the measure of Brownian loops is derived using the Polyakov-Alvarez formula.

Study isotropic solutions in smooth metric measure spaces with vacuum Einstein equations.

We show that the driving force behind the regularizing effect of Laplacian smoothing on surface elements is the popular mean ratio quality measure. We use these insights to provide natural generalizations to polygons and polyhedra. The corresponding functions measuring the quality of meshes are easily seen to be convex…

We first extend Cheeger-Colding Almost Splitting Theorem to smooth metric measure spaces. Arguments utilizing this extension of the Almost Splitting Theorem show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abe…

In this article, we introduce an analogous problem to Yamabe type problem considered by Case, J., which generalizes the Escobar-Riemann mapping problem for smooth metric measure spaces with boundary. The last problem will be called Escobar-Riemann mapping type problem. For this purpose, we consider the generalization o…

We propose a definition of the weighted $σ_k$-curvature of a smooth metric measure space and justify it in two ways. First, we show that the weighted $σ_k$-curvature prescription problem is governed by a fully nonlinear second order elliptic PDE which is variational when $k=1,2$ or the smooth metric measure space is lo…

In this paper, we extend the Hamilton's gradient estimates \cite{har93} and a monotonicity formula of entropy \cite{ni04} for heat flows from smooth Riemannian manifolds to (non-smooth) metric measure spaces with appropriate Riemannian curvature-dimension condition.

We study a transformation of metric measure spaces introduced by Gigli and Mantegazza consisting in replacing the original distance with the length distance induced by the transport distance between heat kernel measures. We study the smoothing effect of this procedure in two important examples. Firstly, we show that in…

Study stationary measures and orbit closures for non-abelian actions on surfaces.

Study KMS measures in Poisson geometry, focusing on $b$-Poisson manifolds.

One way to interpret smoothness of a measure in infinite dimensions is quasi-invariance of the measure under a class of transformations. Usually such settings lack a reference measure such as the Lebesgue or Haar measure, and therefore we can not use smoothness of a density with respect to such a measure. We describe h…

Dynamic angles estimated from noisy measurements over time with smoothness constraints.

We study a "div-grad type" sub-Laplacian with respect to a smooth measure and its associated heat semigroup on a compact equiregular sub-Riemannian manifold. We prove a short time asymptotic expansion of the heat trace up to any order. Our main result holds true for any smooth measure on the manifold, but it has a spec…

Let $(M^{n+1},g,e^{-f}dμ)$ be a complete smooth metric measure space with $2\leq n\leq 6$ and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded $f$-minimal hypersurfaces in $M$ with uniform upper bounds on $f$-index and weighted vo…

Smooth solutions found for a specific type of Yamabe problem.

Study solves Yamabe problems on metric measure spaces with or without boundary.

New algorithm reduces prediction error in online learning without knowing base measure.

Solves $L_p$-Gaussian chord Minkowski problem using Gauss curvature flow.

We prove that two-step analytic sub-Riemannian structures on a compact analytic manifold equipped with a smooth measure and Lipschitz Carnot groups satisfy measure contraction properties.

The paper measures and limits the extent of non-smooth points in Alexandrov spaces.

In this paper we study complete manifolds equipped with smooth measures whose spectrum of the weighted Laplacian has an optimal positive lower bound and the $m$-dimensional Bakry-Émery Ricci curvature is bounded from below by some negative constant. In particular, we prove a splitting type theorem for complete smooth m…

Develops new synthetic Ricci flow concepts for metric measure spaces.

Study shows central limit theorem for counting measures in non-smooth spaces.

A measured solenoid is a laminated space endowed with a tranversal measure invariant by holonomy, as defined in arXiv:0910.2836. A measured solenoid immersed in a smooth manifold produces a closed current (known as generalized Ruelle-Sullivan current). Uniquely ergodic solenoids are those for which there is a unique (u…

We describe and partially solve a natural Yamabe-type problem on smooth metric measure spaces which interpolates between the Yamabe problem and the problem of finding minimizers for Perelman's $ν$-entropy. This problem reduces in all dimensions on Euclidean space to the characterization of the minimizers of the family …

In this paper, we present the optimization formulation of the Kalman filtering and smoothing problems, and use this perspective to develop a variety of extensions and applications. We first formulate classic Kalman smoothing as a least squares problem, highlight special structure, and show that the classic filtering an…

Extends randomized smoothing to certify robustness against various threat models and adversarial perturbations.

Sharp estimates derived for quasilinear equations on metric measure spaces.

Smooth metric measure spaces have been studied from the two different perspectives of Bakry-Émery and Chang-Gursky-Yang, both of which are closely related to work of Perelman on the Ricci flow. These perspectives include a generalization of the Ricci curvature and the associated quasi-Einstein metrics, which include Ei…