arXiv research

Estimates spectral projections restricted to uniformly embedded submanifolds.

Extends flat submanifold properties from hyperbolic plane to symmetric spaces.

In this work we prove that every locally symmetric smooth submanifold gives rise to a naturally defined smooth submanifold of the space of symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to the locally symmetric manifold. We also present an explicit f…

Paper analyzes convergence of Laplacian eigenmaps on submanifolds with singularities.

Study of submanifolds in symplectic and contact manifolds using Hausdorff metrics.

The paper studies essential spectra of submanifolds in Euclidean spaces.

The paper defines a new concept of approximability for Lagrangian submanifolds.

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

Laplacian Eigenvectors of the graph constructed from a data set are used in many spectral manifold learning algorithms such as diffusion maps and spectral clustering. Given a graph constructed from a random sample of a $d$-dimensional compact submanifold $M$ in $\mathbb{R}^D$, we establish the spectral convergence rate…

A new model clusters networks with community-specific submanifold structures.

Proves new inequality linking spectral numbers of Lagrangians and their reductions.

Develops an oblique projection technique to approximate a foliation for non-normal dynamics.

This work connects point particles to spin chains using geometric methods.

In Part I, we develop the notions of a Moebius structure and a conformal Cartan geometry, establish an equivalence between them; we use them in Part II to study submanifolds of conformal manifolds in arbitrary dimension and codimension. We obtain Gauss-Codazzi-Ricci equations and a conformal Bonnet theorem characterizi…

Study of free boundary minimal Möbius bands in spherical caps.

The abstract discusses a spectral sequence for Lie algebroids.

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

We prove spectral, stochastic and mean curvature estimates for complete $m$-submanifolds $\varphi \colon M \to N$ of $n$-manifolds with a pole $N$ in terms of the comparison isoperimetric ratio $I_{m}$ and the extrinsic radius $r_\varphi\leq \infty$. Our proof holds for the bounded case $r_\varphi< \infty$, recovering …

Study shows convergence of Lagrangian submanifolds under certain metrics.

Quantum propagation studied for Berezin-Toeplitz operators.

Sharp spectral estimates for negatively curved foliations.

A membrane technique, in which the symplectic and Ricci forms are integrated over surfaces in a complexification of the phase space, as well a ``creation" connection with zero curvature over lagrangian submanifolds, is used to obtain a unified quantization including a noncommutative algebra of functions, its representa…

Study geometric properties and spectral estimates on warped products.

Study shows infinite Hofer diameter for Lagrangian orbits in cotangent bundles.

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in the setting of Riemannian manifolds of bounded geometry. Bounded geometry of the ambient manifold is a crucial assumption required to control the uniformity of all estimates throughout the proof. The $C^{k,α}$-smoothness result is o…

Study establishes Pólya-Szegő inequalities on submanifolds with small total mean curvature.

Study of unitary and groupoid orbits of normal operators, focusing on manifold structures and spectral conditions.

Study free boundary minimal submanifolds in geodesic balls in hyperbolic and spherical spaces.

In this paper we compute the Reidemeister torsion of a isoenergetic surface for the integrable Hamiltonian system on the four-dimensional symplectic manifold. We use the spectral sequence defined by the filtration and following Witten-Floer ideas we bring into play the orbits connecting the critical submanifolds.

Study of Lagrangian submanifolds with Riemannian bounds and their metric properties.

In this work, we study the pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss map. First, we classify the Lorentzian surfaces in a 4-dimensional pseudo-sphere $\mathbb{S}^4_s(1)$ with index s, $s=1, 2$, and having harmonic pseudo-spherical Gauss map. Then we give a characterization the…

The paper studies the asymptotic expansion of Gaussian integral operators on Riemannian submanifolds.

The paper studies Hodge Laplacians from point clouds, proving spectral convergence and harmonic form consistency.

By only using spectral theory of the Laplace operator on spheres, we prove that the unit 3-dimensional sphere of a 2-dimensional complex subspace of $\mathbb{C}^3$ is a $Ω$-stable submanifold with parallel mean curvature, when $Ω$ is the Kähler calibration of rank 4 of $\mathbb{C}^3$.

Proves nearby Lagrangian cocores are homotopically rigid in certain dimensions.

Magnitude of geometric shapes studied for smooth manifolds, revealing spectral geometry insights.

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…

We prove a number of results relating various measures (volume, Legendrian index, stability index, and spectral curve genus) of the geometric complexity of special Lagrangian $T^2$-cones. We explain how these results fit into a program to understand the "most common" three-dimensional isolated singularities of special …

We prove the existence of a spectral sequence for Lagrangian Floer homology which converges to the Floer homology of the image of a Lagrangian submanifold under multiple fibred Dehn twists. The $E_1$ term of the sequence is given by the hypercube of "resolutions" of the Dehn twists involved. The proof relies on the exa…

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…

Free boundary minimal submanifolds with boundaries on concentric spheres

We study conformal $Spin$-subgeometry of submanifolds in a semi-Riemannian $Spin$-manifold, focusing on conformal $Spin$-manifolds $(M,[h])$ and their Poincaré-Einstein metrics $(X,g_+)$. Our approach is based on the spectral theory of Dirac operator in the ambient $Spin$-manifold, and associated spinor valued meromorp…

We extend the Heegaard Floer homological definition of spectral order for closed contact 3-manifolds due to Kutluhan, Matić, Van Horn-Morris, and Wand to contact 3-manifolds with convex boundary. We show that the order of a codimension zero contact submanifold bounds the order of the ambient manifold from above. As the…

For a transversal pair of closed Lagrangian submanifolds L, L' of a symplectic manifold M so that $π_{1}(L)=π_{1}(L')=0=c_{1}|_{π_{2}(M)}=ω|_{π_{2}(M)}$ and a generic almost complex structure J we construct an invariant with a high homotopical content which consists in the pages of order $\geq 2$ of a spectral sequence…

We discuss $SU(2)$ Bogomolny monopoles of arbitrary charge $k$ invariant under various symmetry groups. The analysis is largely in terms of the spectral curves, the rational maps, and the Nahm equations associated with monopoles. We consider monopoles invariant under inversion in a plane, monopoles with cyclic symmetry…

We establish inequalities for the eigenvalues of Schrödinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, Pólya, and Wei…

We introduce two families of soliton hierarchies: the twisted hierarchies associated to symmetric spaces. The Lax pairs of these two hierarchies are Laurent polynomials in the spectral variable. Our constructions gives a hierarchy of commuting flows for the generalized sine-Gordon equation (GSGE), which is the Gauss-Co…

The 1-d Schrodinger flow on 2-sphere, the Gauss-Codazzi equation for flat Lagrangian submanifolds in C^n, and the space-time monopole equation are all examples of geometric soliton equations. The linear systems with a spectral parameter (Lax pair) associated to these equations satisfy the reality condition associated t…