arXiv research

We show that for elliptic parametric functionals whose Wulff shape is smooth and has strictly positive curvature, any surface with constant anisotropic mean curvature which is a topological sphere is a rescaling of the Wulff shape.

Study shows limits of volume-constrained sets are finite unions of Wulff shapes.

We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems, where the bifurcation parameter is introduced by shrinking the domain to a point. Th…

New DDMs use neural networks for solving equations on manifold shapes.

PEA improves PCA and k-means for non-linear data and complex clusters.

We construct simply connected, minimal, symplectic 4-manifolds with exotic smooth structures and each with one Seiberg-Witten basic class up to sign, on the Noether line and between the Noether and half Noether lines by star surgeries introduced by Karakurt and Starkston, and by using complex singularities. We also con…

Given an elliptic integrand of class $\mathscr{C}^{3}$, we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.

Modeling wildfire spread using Gielis superformula and Finsler spacetime.

New model reduces bias in cosmic shear measurements.

Study compares geometric approaches for shape and deformation statistics.

The study finds the best elliptical trajectory for planets using a variation of the hodograph theorem.

In this paper, we investigate a transition from an elastica to a piece-wised elastica whose connected point defines the hinge angle $φ_0$; we refer the piece-wised elastica $Λ_{φ_0}$-elastica or $Λ$-elastica. The transition appears in the bending beam experiment; we compress elastic beams gradually and then suddenly du…

In this work we study the affine principal lines of surfaces in 3-space. We consider the binary differential equation of the affine curvature lines and obtain the topological models of these curves near the affine umbilic points (elliptic and hyperbolic). We also describe the generic behavior of affine curvature lines …

We find the shape of the Donaldson invariants of a 4-manifold with b_1=0 and b^+>1, which may be not of simple type. The invariants appear as the q^0 coefficient of a expression given in terms of modular forms (as was predicted by Moore and Witten). We re-express the formula using complete elliptic integrals to prove a…

We study the geometry of complete immersed surfaces in $\mathbb{R}^3$ with constant anisotropic mean curvature (CAMC). Assuming that the anisotropic functional is uniformly elliptic, we prove that: (1) planes and CAMC cylinders are the only complete surfaces with CAMC whose Gauss map image is contained in a closed hemi…

Paper proves uniqueness of catenary cylinders based on their asymptotic shape.

The paper proves a conjecture about the shape of floating bodies.

We are concerned with fully nonlinear elliptic equations on complex manifolds and search for technical tools to overcome difficulties in deriving a priori estimates which arise due to the nontrivial torsion and curvature, as well as the general (non-pseudoconvex) shape of the boundary. We present our methods, which wor…

We study the real Monge-Ampère equation in two and three dimensions, both from the point of view of the SYZ conjecture, where solutions give rise to semi-flat Calabi-Yau's and in affine differential geometry, where solutions yield parabolic affine sphere hypersurfaces. We find explicit examples, connect the holomorphic…

In this article we study the class of right-invariant, fractional order Sobolev-type metrics on groups of diffeomorphisms of a compact manifold M. Our main result concerns well-posedness properties for the corresponding Euler-Arnold equations, also called the EPDiff equations, which are of importance in mathematical ph…

A new vine copula mixture model improves clustering accuracy for non-Gaussian data.

Abstract M5 branes on ADE singularities yields BPS spectrum and partition functions.

Constructs Lie groupoid integrating elliptic tangent bundles and Poisson structures.

Study finds negatively curved spheres in elliptic surfaces and their modifications.

In this paper, we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle. Such projective elliptic genera are rational q-series that have topological definition and also have analytic interpretation via the fractional index theorem in…

New method for analyzing elliptic and parabolic equations.

We give a systematic method to calculate some homological data from the global monodromy of a topological elliptic surface. We apply this method to the cases 1) the transcendental lattice of an extremal elliptic K3 surface, 2) the torsion part of Mordell-Weil group of a general elliptic surface, and 3) the Mordell-Weil…

New topological obstructions found for elliptic and quasiregularly elliptic manifolds.

Let (P1) be certain elliptic free-boundary problem on a Riemannian manifold (M,g). In this paper we study the restrictions on the topology and geometry of the fibres (the level sets) of the solutions f to (P1). We give a technique based on certain remarkable property of the fibres (the analytic representation property)…

This study explores the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators using $KK$-theory.

Elliptic bouquets defined for spin manifolds with circular actions.

We give a parameterization of Alfred Gray's Elliptical Catenoid and Elliptical Hellicoid using Jacobi's elliptic functions. This parameterization avoids some problems present in the original depiction of these surfaces.

Study the relationship between canonical polynomials and elliptic sequences for elliptic singularities.

Elliptic Chern characters and Atiyah-Witten formula generalized to double loop spaces.

Mathai, Melrose, and Singer introduced the notion of projective elliptic operators on manifolds equipped with an Azumaya bundle. In this note we compute the equivariant index of transversally elliptic operators that are the pullback of projective elliptic operators on the trivialization of the Azumaya bundle. It encomp…

New deficit functions link elliptic and parabolic inequalities, proving log Sobolev.

New proof for curved 3-cohom manifold rational ellipticity.

Characterizes totally elliptic surface group representations into Lie groups.

A guide for solving first-order elliptic boundary value problems.

Elliptical processes extend Gaussian models with heavier tails.

Introduces spectral-domain Wasserstein distance and Gelbrich bound for elliptical processes.

New Witten rigidity theorems for elliptic genus in various dimensions.

Elliptical processes generalize Gaussian and Student-t models with fat tails and computational efficiency.

Researchers compute the cohomology of an elliptic tangent bundle.

Paper defines invariants for elliptic Weyl groups and connects them to Frobenius structures.

This note shows how independent elliptical distributions minimize the Wasserstein distance.

Study of 17 surface behaviors and singularities for elliptic Weingarten equations.

Geomstats introduces shape module for analyzing shapes of objects.