arXiv research

We prove an almost splitting theorem for the warped product space with warped function $f(r)=\cosh\left(r\sqrt{\fracλ{n-2}}\right)$.

Unified proof of smooth fibration theorems for collapsed manifolds.

By Cheeger-Colding's almost splitting theorem, if a domain in a Ricci flat manifold is pointed-Gromov-Hausdorff close to a lower dimensional Euclidean domain, then there is a harmonic almost splitting map. We show that any eigenfunction of the Laplace operator is almost constant along the fibers of the almost splitting…

New proof linking scalar curvature to volume growth on 3-manifolds.

Proves effective linear volume growth for 3-manifolds with positive scalar curvature.

We first extend Cheeger-Colding Almost Splitting Theorem to smooth metric measure spaces. Arguments utilizing this extension of the Almost Splitting Theorem show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abe…

Study almost rigidity of super Ricci flow with non-negative Muller quantity.

Study of splitting maps in Type I Ricci flows for understanding singular set structure.

The study shows black hole horizons at low temperatures have limited topology.

For any complete $n$-dim Riemannian manifold $M^n$ with nonnegative Ricci curvature, Kapovitch and Wilking proved that any finitely generated subgroup of the fundamental group $π_1(M^n)$ can be generated by $C(n)$ generators. Inspired by their work, we give a quantitative proof of the above theorem and show that $C(n)\…

Stability of positive mass theorem proven under Ricci curvature bounds.

We prove a global Birkhoff decomposition for almost split real forms of loop groups, when an underlying finite dimensional Lie group is compact. Among applications, this shows that the dressing action - by the whole subgroup of loops which extend holomorphically to the exterior disc - on the $U$-hierarchy of the ZS-AKN…

We prove that a metric measure space $(X,d,m)$ satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space $W^{1,2}$ is Hilbert is rectifiable. That is, a $RCD^*(K,N)$-space is rectifiable, and in particular for $m$-a.e. point the tangent cone is unique and euclidean of dimension at most $N$. The…

The universe's shape and size are determined in general cosmological models.

Stability of timelike Ricci bounds in low-regularity spacetimes.

We describe some buildings related to complex Kac-Moody groups. First we describe the spherical building of SLn(C) (i.e. the projective geometry PG(Cn)) and its Veronese representation. Next we recall the construction of the affine building associated to a discrete valuation on the rational function field $C(z)$. Then …

Solves the gauge problem for Ricci flow cylinders, proving strong rigidity.

Polynomial growth bounds for eigenfunctions on non-compact spaces.

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.