arXiv research

Support theorem proved for X-ray transform on certain non-compact manifolds.

A support theorem for the horocycle Radon transform f \to \hat{f} is a property of the form \hat{f} of compact support \rightarrow f of compact support. Here we prove a variation of this result where support (\hat{f}) is outside a fixed horocycle in hyperbolic space.

Support theorem for tensor fields on Riemannian manifolds.

In this work, we show an injectivity result and support theorems for integral moments of a m-tensor field on a simple, real analytic, Riemannian manifold. Integral moments of m-tensor field were first introduced by Sharafutdinov. At first we generalize a Helgason type support theorem proven by Krishnan and Stefanov in …

We study the weighted ray transform of integrating functions on a Lorentzian manifold over lightlike geodesics. We prove support theorems if the manifold and the weight are analytic.

For a convex domain $D$ that is enclosed by the hypersurface $\partial D$ of bounded normal curvature, we prove an angle comparison theorem for angles between $\partial D$ and geodesic rays starting from some fixed point in $D$, and the corresponding angles for hypersurfaces of constant normal curvature. Also, we obtai…

A geometric version of the Poincaré Lemma is established for the topological vector space of differential chains. In particular, every differential k-cycle with compact support in a contractible open subset U of a smooth n-manifold M is the boundary of a differential (k+1) -chain with compact support in U. Applications…

Support selection and eventwise decoupling for simultaneous bets proven.

We extend the Besicovitch-Federer projection theorem to transversal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural 2-dimensional measures invariant under the geodesic flow having 2-dimensional supports s…

Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.

Extends Polydisk Theorem to Cartan-Hartogs domains.

The paper proves rigidity results for capillary hypersurfaces in hyperbolic space.

We construct the coarse index class with support condition (as an element of coarse $K$-homology) of an equivariant Dirac operator on a complete Riemannian manifold endowed with a proper, isometric action of a group. We further show a coarse relative index theorem and discuss the compatibility of the index with the sus…

Supports conjecture about harmonic maps from S³ to S².

X-ray transform on H-type groups solved, revealing function injectivity.

In this paper we prove a universal inequality describing the asymptotic behavior of support points for planar continuous curves. As corollaries we get an analogous result for tangent points of differentiable planar curves and some (partially known) assertions on the asymptotic of the mean value points for various class…

Paper studies a new curvature system and proves rigidity and gap theorems.

In this paper we prove a new inversion theorem and a refinement of an old support theorem for two Radon transforms on a symmetric space. Included are some new identities for the Abel transform and some results about the Fourier transform from a joint work with Rawat, Sengupta and Sitaram.

The paper defines positivity for singular metrics on vector bundles and proves related theorems.

Given a normed plane $\mathcal{P}$, we call $\mathcal{P}$-cycloids the planar curves which are homothetic to their double $\mathcal{P}$-evolutes. It turns out that the radius of curvature and the support function of a $\mathcal{P}$-cycloid satisfy a differential equation of Sturm-Liouville type. By studying this equati…

In this article, we initiate a geometric measure theoretic approach to symplectic Hodge theory. In particular, we apply one of the central results in geometric measure theory, the Federer-Fleming deformation theorem, together with the cohomology theory of normal cur- rents on a differential manifold, to establish a fun…

The paper studies deformations of Lagrangian submanifolds using algebraic tools.

By adapting the Cheeger-Simons approach to differential cohomology, we establish a notion of differential cohomology with compact support. We show that it is functorial with respect to open embeddings and that it fits into a natural diagram of exact sequences which compare it to compactly supported singular cohomology …

Central limit theorem for Green metrics on hyperbolic groups.

We use SO(3) gauge theory to define a functor from a category of unoriented webs and foams to the category of finite-dimensional vector spaces over the field of two elements. We prove a non-vanishing theorem for this SO(3) instanton homology of webs, using Gabai's sutured manifold theory. It is hoped that the non-vanis…

We prove a Paley-Wiener Theorem for a class of symmetric spaces of the compact type, in which all root multiplicities are even. This theorem characterizes functions of small support in terms of holomorphic extendability and exponential type of their (discrete) Fourier transforms. We also provide three independent new p…

We provide a somewhat geometric proof of a rigidity theorem by M. Ledoux and C. Xia concerning complete manifolds with non-negative Ricci curvature supporting an Euclidean-type Sobolev inequality with (almost) best Sobolev constant. Using the same technique we also generalize Ledoux-Xia result to complete manifolds wit…

Researchers develop a method to infer reference measures from observed functionals.

The paper analyzes sparse PCA for incomplete data and proves support recovery conditions.

Vanishing result for cohomology leads to extension theorem for pluriharmonic functions.

Study of spinors and tangent groupoid for index theorem.

Let $(M,g)$ be a simple Riemannian manifold with boundary and consider the geodesic ray transform of symmetric 2-tensor fields. Let the integral of $f$ along maximal geodesics vanish on an appropriate open subset of the space of geodesics in $M$. Under the assumption that the metric $g$ is real-analytic, it is shown th…

Paper extends cohomology classes and holomorphic sections on subvarieties.

Paper establishes comparison theorems for large-margin learning.

Develops support theorem for analytic transforms in tomography.

The braneworld theory appear with the purpose of solving the problem of the hierarchy of the fundamental interactions. The perspectives of the theory emerge as a new physics, for example, deviation of the law of Newton's gravity. One of the principles of the theory is to suppose that the braneworld is local submanifold…

We give a shorter proof of the following theorem of Kathryn Mann \cite{M}: the identity component of the group of the compactly supported $C^r$ diffeomorphisms of $\R^n$ cannot admit a nontrivial $C^p$-action on $S^1$, provided $n\geq2$, $r\neq n+1$ and $p\geq2$. We also give a new proof of another theorem of Mann: any…

Study risk-constrained Kelly optimization for mutually exclusive outcomes, proving support invariance and developing a structured algorithm.

L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of a locally symmetric space. We define the micro-support of an L-module; it is a set of irreducible modules for the Levi quotients of the parabolic Q-subgroups associated to the strata. We prove a vanishing th…

The paper extends Thurston's method to new variants of Mather-Thurston theorem.

We show that a transverse link in a contact structure supported by an open book decomposition can be transversely braided. We also generalize Markov's theorem on when the closures of two braids represent (transversely) isotopic links.

The paper proves a Hard Lefschetz Theorem for push-forwards of polarized twistor modules.

The paper extends sphere theorems to higher-order mean curvature functions on specific hypersurfaces.

Given a geodesic inside a simply-connected, complete, non-positively curved Riemannian (NPCR) manifold M, we get an associated geodesic inside the asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is contained inside a bi-Lipschitz flat, then the original geodesic supports a non-trivial, orthog…

We develop the fundamental theorem of asset pricing in a probability-free infinite-dimensional setup. We replace the usual assumption of a prior probability by a certain continuity property in the state variable. Probabilities enter then endogenously as full support martingale measures (instead of equivalent martingale…

Study cohomology of generalized configuration spaces on topological spaces.

We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic $p$-Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconform…

We give an alternative proof of a theorem of Honda-Kazez-Matić that every non-right-veering open book supports an overtwisted contact structure. We also study two types of examples that show how overtwisted discs are embedded relative to right-veering open books.