arXiv research

Explains weightings along submanifolds, focusing on Lie groupoids.

Upper bounds for eigenvalues on submanifolds in weighted manifolds.

The study explores weightings on submanifolds and their geometric properties.

In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non zero eigenvalue of the drift Lapl…

The paper studies variations of weighted curvature on submanifolds.

The paper proves a Wulff inequality for minimal submanifolds with boundary in Euclidean space.

In this paper, we prove a Sobolev and isoperimetric inequalities for submanifold in weighted manifold. Our results generalize the Hoffman-Spruck's inequalities.

In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if…

Study of weighted nonlinear flags in symplectic geometry.

Study finds formulas for minimal submanifolds using Möbius transformations.

Upper bounds found for eigenvalues of weighted Steklov and (p,q)-Laplacian problems.

In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on "convenient" vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold $(M,ω)$, we construct a weak symplectic structure on each leaf ${\textbf I}_{w}$ of a foli…

In this paper we generalize examples of Hamiltonian stationary Lagrangian submanifolds constructed by Lee and Wang in $\mathbb{C}^m$ to toric almost Calabi-Yau manifolds. We construct examples of weighted Hamiltonian stationary Lagrangian submanifolds in toric almost Calabi-Yau manifolds and solutions of generalized La…

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, …

The study of quotient structures in multi-graded bundles, including double vector bundles.

The paper proves cohomology vanishing for a specific type of minimal submanifolds in a weighted Euclidean ball.

In this paper, we investigate submanifolds with locally bounded mean curvature in Hadamard manifolds, product manifolds $N\times\mathbb{R}$, submanifolds with bounded $\varphi$-mean curvature in the hyperbolic space, and successfully give lower bounds for the weighted fundamental tone and the first eigenvalue of the $p…

Let $P$ be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight $e^h$. The aim of this paper is twofold. First, by assuming certain control on the $h$-mean curvature of $P$, we establish comparisons for the $h$-capacity of extrinsic balls in $P$, from which we ded…

Develops methods for computing conformal invariants of submanifolds.

Sharp Hardy inequalities on Riemannian submanifolds with non-negative curvature.

A conformal structure on a manifold $M^n$ induces natural second order conformally invariant operators, called Möbius and Laplace structures, acting on specific weight bundles of $M$, provided that $n\ge 3$. By extending the notions of Möbius and Laplace structures to the case of surfaces and curves, we develop here th…

The aim of this paper is to prove isoperimetric inequalities on submanifolds of the Euclidean space using mass transportation methods. We obtain a sharp ?weighted isoperimetric inequality? and a nonsharp classical inequality similar to the one obtained by J. Michael and L. Simon. The proof relies on the description of …

After works by Michael and Simon [10], Hoffman and Spruck [9], and White [14], the celebrated Sobolev inequality could be extended to submanifolds in a huge class of Riemannian manifolds. The universal constant obtained depends only on the dimension of the submanifold. A sort of applications to the submanifold theory a…

The paper classifies hypersurfaces with constant weighted mean curvature.

This paper develops a method to learn lower-dimensional submanifolds of brain connectomes.

The paper validates Stokes' theorem for differential subcomplexes in positively graded Lie groups.

$ξ$-submanifold in the Euclidean space $\bbr^{m+p}$ is a natural extension of the concept of self-shrinker to the mean curvature flow in $\bbr^{m+p}$. It is also a generalization of the $λ$-hypersurface defined by Q.-M. Cheng et al to arbitrary codimensions. In this paper, some characterizations for $ξ$-submanifolds ar…

We prove a spanning result for vector-valued Poincaré series on a bounded symmetric domain. We associate a sequence of holomorphic automorphic forms to a submanifold of the domain. When the domain is the unit ball in ${\Bbb{C}}^n$, we provide estimates for the norms of these automorphic forms and we find asymptotics of…

Study eigenvalues of a generalized p-Laplacian on forms.

New inequalities for submanifolds in curved spaces.

Let $(M,g)$ be a complete non-compact Riemannian manifold together with a function $e^h$, which weights the Hausdorff measures associated to the Riemannian metric. In this work we assume lower or upper radial bounds on some weighted or unweighted curvatures of $M$ to deduce comparisons for the weighted isoperimetric qu…

New collapsing mechanism for G2-manifolds discovered.

Tropical curves match to special Lagrangian shapes.

In this paper we study the asymptotic behavior of second-order uniformly elliptic operators on weighted Riemannian manifolds. They naturally emerge when studying spectral properties of the Laplace-Beltrami operator on families of manifolds with rapidly oscillating metrics. We appeal to the notion of H-convergence intro…

We consider a connected symplectic manifold $M$ acted on properly and in a Hamiltonian fashion by a connected Lie group $G$. Inspired to the recent paper \cite{gb2}, see also \cite{ch} and \cite{pacini}, we study Lagrangian orbits of Hamiltonian actions. The dimension of the moduli space of the Lagrangian orbits is giv…

We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a $m$-dimensional submanifold $M$ in $R^d$ as the sample size $n$ increases and the neighborhood size $h$ tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of…

Free boundary minimal submanifolds with boundaries on concentric spheres

We prove a generalization of Hsiung-Minkowski formulas for closed submanifolds in semi-Riemannian manifolds with constant curvature. As a corollary, we obtain volume and area upper bounds for k-convex hypersurfaces in terms of a weighted total k-th mean curvature of the hypersurface. We also obtain some Alexandrov-type…

Develops theory of weightings for Lie groupoids and algebroids.

A visible action on a complex manifold is a holomorphic action that admits a $J$-transversal totally real submanifold $S$. It is said to be strongly visible if there exists an orbit-preserving anti-holomorphic diffeomorphism $σ$ such that $σ|_S = \mathrm{id}$. In this paper, we prove that for any Hermitian symmetric sp…

We study forgetful maps between Deligne-Mostow moduli spaces of weighted points on P^1, and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livné fibrations and the Mostow/Toledo maps between complex hyperbolic surfaces. They also in…

The paper develops manifold learning in Wasserstein space for probability measures.

Derives integral formula for differential forms on compact spaces with applications.

The paper generalizes Alexandrov theorems for null hypersurfaces with integral curvature conditions.

We generalize Fulton and MacPherson's configuration space construction to weighted filtered manifolds.

The paper proves weighted monotonicity theorems in various spaces and applies them to minimal surfaces.

We consider odd Laplace operators arising in odd symplectic geometry. Approach based on semidensities (densities of weight 1/2) is developed. The role of semidensities in the Batalin--Vilkovisky formalism is explained. In particular, we study the relations between semidensities on an odd symplectic supermanifold and di…

Study circle actions on manifolds with 3 fixed points, finding dimension constraints and unique structures.