arXiv research

This paper analyzes the Bochner formula for Riemannian flows and derives eigenvalue estimates.

In [4], we gave a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. In this paper, we extend this result to the case of the first eigenvalue on basic $p$-forms for $p>1$. As in [4], the limiting cas…

In this paper, we give a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. The limiting case gives rise to a particular geometry of the flow and the boundary. Namely, the flow is a local product and…

Study wave invariants for Riemannian foliations, showing independence of mean curvature.

We obtain a basic inequality involving the Laplacian of the warping function and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. Applying this general theory, we obtain basic inequa…

The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general form of a Laplacian on …

Compact polyhedral surfaces (or, equivalently, compact Riemann surfaces with conformal flat conical metrics) of an arbitrary genus are considered. After giving a short self-contained survey of their basic spectral properties, we study the zeta-regularized determinant of the Laplacian as a functional on the moduli space…

In this article we present an intrinsec construction of foliated Brownian motion via stochastic calculus adapted to foliation. The stochastic approach together with a proposed foliated vector calculus provide a natural method to work on harmonic measures. Other results include a decomposition of the Laplacian in terms …

In this paper, we prove the invariance of the spectrum of the basic Dirac operator defined on a Riemannian foliation $(M,\mathcal{F})$ with respect to a change of bundle-like metric. We then establish new estimates for its eigenvalues on spin flows in terms of the O'Neill tensor and the first eigenvalue of the Dirac op…

The paper surveys the Guillemin-Uribe trace formula for magnetic Laplacians.

Dirichlet-Neumann duality for Riemannian submersions

Study shows exponential decay of Bergman kernels on manifolds with bounded Ricci curvature.

Study of coupled Sasaki-Einstein and solitons metrics.

We study the regularized determinant of the Laplacian as a functional on the space of Mandelstam diagrams (noncompact translation surfaces glued from finite and semi-infinite cylinders). A Mandelstam diagram can be considered as a compact Riemann surface equipped with a conformal flat singular metric $|ω|^2$, where $ω$…

We study space-like self-shrinkers of dimension $n$ in pseudo-Euclidean space $\ir{m+n}_m$with index $m$. We derive drift Laplacian of the basic geometric quantities and obtain their volume estimates in pseudo-distance function. Finally, we prove a rigidity results under minor growth conditions interms of the mean curv…

The paper investigates exotic smooth structures on manifolds with group actions.

We consider the basic heat operator on functions on a Riemannian foliation of a compact, Riemannian manifold, and we show that the trace of this operator has a particular short time asymptotic expansion. The coefficients in this expansion are obtainable from local transverse geometric invariants - functions computable …

The study uses unsupervised machine learning to identify top European football teams.

The problem of ranking a set of objects given some measure of similarity is one of the most basic in machine learning. Recently Agarwal proposed a method based on techniques in semi-supervised learning utilizing the graph Laplacian. In this work we consider a novel application of this technique to ranking binary choice…

This is a paper about geometry of (iterated) variations. We explain why no sources of divergence are built into the Batalin-Vilkovisky (BV) Laplacian, whence there is no need to postulate any ad hoc conventions such as "$δ(0)=0$" and "$\logδ(0)=0$" within BV-approach to quantisation of gauge systems. Remarkably, the ge…

Let M be a closed compact n-dimensional manifold with n odd. We calculate the first and second variations of the zeta-regularized determinants det^\primeΛand det L as the metric on M varies, where Δdenotes the Laplacian on functions and L denotes the conformal Laplacian. We see that the behavior of these functionals de…

Study on deformations of Einstein and nearly G2 structures in 3-Sasaki manifolds.

The paper compares isoperimetric quotients and capacities in weighted manifolds.

Efficiently fits stratified models with Laplacian regularization for categorical features.

Let F be a riemannian flow on a closed manifold M. We study the behavior of the first eigenvalues of the Hodge Laplacian acting on differential forms under adiabatic collapsing of the flow. We show that the number of small eigenvalues is related to the basic cohomology of F, and give spectral criteria for the vanishing…

New CNNs for set functions using powerset shifts.

Einstein's non-symmetric geometry uses Bochner's technique to prove decomposition and vanishing results.

Defines vector Laplacian on statistical manifolds.

BIG Laplacians bridge combinatorial and Hodge Laplacians for discrete data.

Paper introduces magnetic Hodge Laplacian for differential forms.

Paper generalizes graph Laplacian to hypergraphs for semi-supervised learning.

The paper extends Laplacian spectra approximations to vector bundles.

Proves Laplacian and Lichnerowicz Laplacian are sectorial in weighted Hölder spaces.

We relate some basic constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular at…

Survey of Laplacian-based methods for data dimensionality reduction and embedding.

The paper sets up eigenvalue comparison theorems for specific Laplacians on manifolds.

Extended bounds on small eigenvalues for pseudo-Laplacians on hyperbolic surfaces.

Researchers solve Laplacian coflow on specific Lie groups.

Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.

Study Hodge Laplacians for manifold data, improving error bounds.

A kinematic method selects the deformation Laplacian for fluid dynamics on Riemannian manifolds.

Root Laplacian Eigenmaps help in spectral embedding of graphs.

New perspective on G2-structures flow from DeTurck Laplacian.

Study solves sub-Laplacian equivalence on a specific Heisenberg group.

Explains BV Laplacian on half-densities in simple terms.

New examples of Laplacian solitons found on solvable Lie groups.

Study shows rates for Laplacian-eigenmap methods in nonparametric regression.

Study improves Reilly's inequality for $p$-Laplacian on submanifolds.