arXiv research

Paper uses 3-circle theorem to study Willmore surfaces and prove decay estimates.

Let $(M; g)$ be a smooth compact Riemiannian manifold without boundary and $g_{k}$ be a metric conformal to $g$. Suppose $vol(M; g_{k})+||R_{k}||_{L^{p}(M;g_{k})} < C$, where $R_{k}$ is the scalar curvature and $p > \frac{n}{2}$. We will use the 3-circle theorem and the John-Nirenberg inequality to study the bubble tre…

Study compact Willmore surfaces without complex structure convergence, computing energy loss and geodesic lengths.

Let $Y\to X$ be a finite normal cover of a wedge of $n\geq 3$ circles. We prove that for any $v\neq 0\in H_1(Y;\mathbb{Q})$ there exists a lift $\widetilde{F}$ to $Y$ of a homotopy equivalence $F:X\to X$ so that the set of iterates $\{\widetilde{F}^d(v): d\in \mathbb{Z}\}\subseteq H_1(Y;\mathbb{Q})$ is infinite. The ma…

Venn diagrams are a graphical way to represent a set system. Each of the n sets is represented by a simple closed curve. The n curves subdivide the plane into 2^n open connected regions, each of which represents the intersection of its containing curves' sets. For example, two overlapping circles can divide the plane i…

AI generates theorems and proofs for training theorem provers.

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.

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.

The study establishes comparison theorems for weighted Finsler manifolds and spacetimes.

Extends three circle theorem to almost Hermitian manifolds.

This is the second of three papers about the Compression Theorem. We give proofs of Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer--Verlag (1986); 2.4.5 C'] and of the Normal Deformation Theorem [The compression theorem I; 4.7], arxiv:math.GT/9712235.

The Bakry-Émery-Ricci tensor is extended and comparison theorems are proven.

Theorem converse to Jordan's curve theorem says that {\it if a compact set $K$ has two complementary domains in $R^{2}$, from each of which it is at every point accessible, it is a simple closed curve}. We show that the requirement of this theorem that {\it all} points of $K$ were accessible from {\it both} complementa…

Survey on mean curvature flow with sphere theorems and Yau rigidity theory.

In the paper titled "Bockstein basis and resolution theorems in extension theory" (arXiv:0907.0491v2), we stated a theorem that we claimed to be a generalization of the Edwards-Walsh resolution theorem. The goal of this note is to show that the main theorem from (arXiv:0907.0491v2) is in fact equivalent to the Edwards-…

New theorem for nonlocal minimal surfaces in any dimension.

We prove Hessian comparison theorems, Laplacian comparison theorems and volume comparison theorems of Finsler manifolds under various curvature conditions. As applications, we derive Mckean type theorems for the first eigenvalue of Finsler manifolds, as well as generalize a result on fundamental group due to Milnor to …

We give exposition of a Liouville theorem established in \cite{Li3} which is a novel extension of the classical Liouville theorem for harmonic functions. To illustrate some ideas of the proof of the Liouville theorem, we present a new proof of the classical Liouville theorem for harmonic functions. Applications of the …