arXiv research

New weighted surface area measures for convex bodies with applications.

Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.

Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.

The CR Obata theorem is extended to weighted Sasakian manifolds.

Synthetic Ricci flows defined for metric measure spaces.

New method calculates Ricci curvature from distances between weighted volumes.

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…

The study examines stable regions in weighted manifolds with boundary properties.

Paper generalizes CR Obata theorem to weighted Sasakian manifolds.

We study geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry-Emery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtai…

Let $(M,\bar{g}, e^{-f}dμ)$ be a complete metric measure space with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove that, in $M$, there is no complete two-sided $L_f$-stable immersed $f$-minimal hypersurface with finite weighted volume. Further, if $M$ is a 3-manifold, we prove a smooth compa…

Defines and proves properties of weighted renormalized volume coefficients.

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Gagliardo-Nirenberg inequality with the same exponent $n$ $(n\geq 2)$, then it has exactly the $n$-dimensional volume growth. Besides, two interesting applications have also been given. The one is that we show that if…

In this paper, we generalize several results for the Hamiltonian stability and the mean curvature flow of Lagrangian submanifolds in a Kähler-Einstein manifold to more general Kähler manifolds including a Fano manifold equipped with a Kähler form $ω\in 2πc_1(M)$ by using the methodology proposed by T. Behrndt. Namely, …

In this paper we prove that on a complete smooth metric measure space with non-negative Bakry-Émery-Ricci curvature if the space of weighted L^2 harmonic one-forms is non-trivial then the weighted volume of the manifold is finite and universal cover of the manifold splits isometrically as the product of the real line w…

New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.

The paper proves a Harnack inequality for heat equations on Finsler metric measure manifolds.

Let $Δ_\varphi = Δ-\nabla \varphi \nabla$ be a symmetric diffusion operator with an invariant weighted volume measure $dμ= e^{-\varphi} dv$ on an $n$-dimensional compact Riemannian manifold $(M,g)$, where $g=g(t)$ solves the extended Ricci flow. In this article we study the evolution and monotonicty of the first nonzer…

Weil-Petersson volumes vary continuously with weighted points on a projective line.

The paper classifies hypersurfaces with constant weighted mean curvature.

Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, we prove a topological splitting theorem of Cheeger-Gromoll type. As a corollary, we prove an isometric splitting theorem for Riemannian manifolds with singularities of nonnegative (Bakry-Emery…

Proposes volumization for neural networks to control bias-variance tradeoff.

In this paper, we first introduce the weighted forward reduced volume of Ricci flow. The weighted forward reduced volume, which related to expanders of Ricci flow, is well-defined on noncompact manifolds and monotone non-increasing under Ricci flow. Moreover, we show that, just the same as the Perelman's reduced volume…

The paper proves geometric inequalities for hypersurfaces in weighted manifolds.

Study submanifolds in gradient Ricci solitons with bounded curvature, proving volume growth properties.

The curvature-dimension condition implies a new weighted scalar curvature.

We prove that a Ricci curvature based method of triangulation of compact Riemannian manifolds, due to Grove and Petersen, extends to the context of weighted Riemannian manifolds and more general metric measure spaces. In both cases the role of the lower bound on Ricci curvature is replaced by the curvature-dimension co…

Sharp isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature.

Let $M$ be a weighted manifold with boundary $\partial M$, i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of the interior weighted area for deformations by hypersurfaces with boundary in $\p…

We study the statistical properties of the recurrence intervals $τ$ between successive trading volumes exceeding a certain threshold $q$. The recurrence interval analysis is carried out for the 20 liquid Chinese stocks covering a period from January 2000 to May 2009, and two Chinese indices from January 2003 to April 2…

New methods for computing volumes and constructing Fano fibrations.

New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.

The paper proves a Laplacian comparison theorem on weighted Riemannian manifolds and applies it to diffusion processes.

Paper estimates Wasserstein distance for Ricci shrinkers.

Let $Ω$ be an open half-space or slab in $\mathbb{R}^{n+1}$ endowed with a perturbation of the Gaussian measure of the form $f(p):=\exp(ω(p)-c|p|^2)$, where $c>0$ and $ω$ is a smooth concave function depending only on the signed distance from the linear hyperplane parallel to $\partialΩ$. In this work we follow a varia…

Researchers compute limits of Kähler-Einstein forms on degenerating manifolds.

Study weakly weighted Einstein-Finsler metrics, showing specific curvature properties and characterizing them.

Bilateral trade relationships in the international level between pairs of countries in the world give rise to the notion of the International Trade Network (ITN). This network has attracted the attention of network researchers as it serves as an excellent example of the weighted networks, the link weight being defined …

Uniqueness found for elliptic equations with drift on manifolds.

Proves stability of cone-volume measure with nearly constant density.

The paper confirms a conjecture linking link bipyramid volume and Mahler measure.

We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype {equation*} \partial_tu= -\sum_{i=1}^{m}X_i^\ast (|\X u|^{p-2} X_i u){equation*} where $p\ge 2$, $ \ \X = (X_1,..., X_m)$ is a system of Lipschitz vector fields defined on a smooth manifold $\M$ endowed with a Borel measure $μ$, …

In this paper, we introduce a definition of $λ$-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that $λ$-hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete $λ$-hypersurfaces with …

Defines magnitude for length spaces with measures, agreeing with finite spaces' magnitude.

Study on manifolds with density using modified Hessians for curvature comparison.

In this paper, we prove that a noncompact complete hypersurface with finite weighted volume, weighted mean curvature vector bounded in norm, and isometrically immersed in a complete weighted manifold is proper. In addition, we obtain an estimate for $f$-stability index of a constant weighted mean curvature hypersurface…

The study proves topological rigidity for certain geometric shapes using Poincaré inequalities.

The study establishes inequalities on Finsler manifolds with weighted Ricci curvature.