arXiv research

New theorem for nonlocal minimal surfaces in any dimension.

New non-existence results for harmonic maps into perturbed cones.

The aim of this work is to extend the results of S. Nayatani about the index and the nullity of the Gauss map of the Costa-Hoffman-Meeks surfaces for values of the genus bigger than 37. That allows us to state that these minimal surfaces are non degenerate for all the values of the genus in the sense of the definition …

We construct higher genus Riemann's minimal surfaces properly embedded in the Euclidean space. To do that we glue end by end a Costa-Hoffman-Meeks examples to two halves genus zero Riemann's minimal surfaces. In first we need to perform a deformation of a Costa-Hoffman-Meeks example to prescribe the flux vector along t…

We present a new construction of embedded minimal surfaces in hyperbolic space with $3$ asymptotically totally geodesic ends and arbitrary finite genus.

The paper constructs surfaces of high genus with three ends.

In this short paper we extend the classical Hoffman-Meeks Halfspace Theorem to self-shrinkers, that is: "Let $P$ be a hyperplane passing through the origin. The only properly immersed self-shrinker $Σ$ contained in one of the closed half-space determined by $P$ is $Σ= P$." Our proof is geometric and uses a catenoid ty…

While it is well known from examples that no interesting `halfspace theorem' holds for properly immersed complete $n$-dimensional self-translating mean curvature flow solitons in Euclidean space $\mathbb{R}^{n+1}$, we show that they must all obey a general `bi-halfspace theorem': Two transverse vertical halfspaces can …

We desingularise the union of $3$ Grim paraboloids along Costa-Hoffman-Meeks surfaces in order to obtain complete embedded translating solitons of the mean curvature flow with $3$ ends and arbitrary finite genus.

Study proves no minimal surfaces can be contained in certain half-spaces or cones.

We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disk. The surfaces are constructed by singular perturbation methods and have three boundary components. They are …

In 1996 M. Traizet obtained singly periodic minimal surfaces with Scherk ends of arbitrary genus by desingularizing a set of vertical planes at their intersections. However, in Traizet's work it is not allowed that three or more planes intersect at the same line. In our paper, by a {\it saddle-tower} we call the desing…

New maxfaces with catenoid or planar ends constructed using node-opening technique.

Study of minimal surfaces in 4D with specific ends.

We consider minimal surfaces $M$ which are complete, embedded and have finite total curvature in $\R^3$, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation $Δu + f(u) = 0 \hbox{in} \R^3$. Here $f=-W'$ with $W$ bistable and balanced, for instance $W(u) =\frac 14 (1-u^2)^2$. We assume that …

Global inverse function theorem proved easily using Riemannian geometry.

A new comparison theorem for geometric spaces.

Paper develops formulas and theorems in Hermitian geometry.

The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.

Revises a theorem by Thurston, finding a counter-example and a weaker version.

Proofs for Moon's theorem and its generalization.

We consider the task of automated theorem proving, a key AI task. Deep learning has shown promise for training theorem provers, but there are limited human-written theorems and proofs available for supervised learning. To address this limitation, we propose to learn a neural generator that automatically synthesizes the…

Analyzes Saito vanishing theorem using $L^{2}$ methods.

Investigates proving geometric theorems over complex and real numbers using tilings.

Extends symplectic reduction and theorem to Lie algebroids.

Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.

Proves Thurston's bounded image theorem for Haken manifolds.

Method upgrades limit theorems to mixing limit theorems for dynamical systems.

Formulates Index III lemma and Rauch III theorem with applications.

In LM, we proved a family version of the famous Witten rigidity theorems and several family vanishing theorems for elliptic genera. In this paper, we gerenalize our theorems LM in two directions. First we establish a family rigidity theorem for the Dirac operator on loop space twisted by general positive energy loop gr…

The paper explains the topological origin of the distinction between incidence theorems over division rings and fields.

Proves two theorems on odd-dimensional manifolds with boundary.

Sharp convergence theorem for sphere submanifolds proved.

INT benchmark tests theorem proving agents' ability to generalize to unseen theorems.

A homological selection theorem for C-spaces, as well as, a finite-dimensional homological selection theorem is established. We apply the finite-dimensional homological selection theorem to obtain fixed-point theorems for usco homologically UV^n set-valued maps.

Abstracts a theorem for non-smooth maps in infinite dimensions.

Atiyah-Singer theorem links math fields, predicts topological insights.

Paper generalizes a theorem for real analytic singularities.

The paper proves injectivity and vanishing theorems on compact Kahler manifolds.

Several proofs of Fáry--Milnor theorem are presented.

Reidemeister's theorem proved using smooth functions and transversality.

Proves an analytic Bertini theorem, generalizing previous work.

This note explores comparison geometry concepts and theorems.

Proof of Tait-Kneser theorem and related variations using Lorentzian geometry.

Proves Skoda's Division Theorem using degeneration and positivity of direct image bundles.

We show how Latour's theorem can be understood as a natural generalization of the $s$-cobordism theorem for cohomology classes $u\in H^1(M;\mathbb{R})$. The $s$-cobordism theorem becomes a special degenerate case when $u=0$.

Generalizes symplectic reduction to cosymplectic groupoid actions.

Strong Frankel theorem for shrinkers in all dimensions.