arXiv research

Optimizes biharmonic map regularity using stratification methods.

The paper improves estimates on singular sets in manifolds with integral curvature bounds.

Extends space theory to singular spaces.

Study curvature growth in 4D singularity models using Perelman's method.

Unified proof of smooth fibration theorems for collapsed manifolds.

Study controls curvature in Ricci flows using necks.

The paper constructs manifolds with infinite holes from a given manifold.

Paper reconciles different Ricci flow approaches and proves weak solutions.

3D Ricci flows have bounded diameter before Type I singularities.

We show that a shrinking Ricci soliton with positive sectional curvature must be compact. This extends a result of Perelman in dimension three and improves a result of Naber in dimension four, respectively.

Sharp Gaussian isoperimetry proven along Ricci flow.

This is a survey article with a limited list of references (as required by the publisher) which appears in the Encyclopedia of Mathematical Physics, eds. J.-P. Francoise, G.L. Naber and Tsou S.T. Oxford: Elsevier, 2006. vol.4, pp.94--104.

The paper extends regularity for $p$-minimizing maps using a Reifenberg Theorem.

Study on energy-minimizing structures in complex geometry.

Study Brownian motion on Perelman's almost Ricci-flat manifold, proving convergence to Ricci flow limits.

We show that if $X$ is a limit of $n$-dimensional Riemannian manifolds with Ricci curvature bounded below and $γ$ is a limit geodesic in $X$ then along the interior of $γ$ same scale measure metric tangent cones $T_{γ(t)}X$ are Hölder continuous with respect to measured Gromov-Hausdorff topology and have the same dimen…

New Calabi-Yau metrics found on C^n for n>=3.

In arXiv:1005.3255 we proved an orbifold Cheeger-Gromov compactness theorem for complete 4d Ricci shrinkers with a lower bound for the entropy, an upper bound for the Euler characterisic, and a lower bound for the gradient of the potential at large distances. In this note, we show that the last two assumptions in fact …

The paper characterizes upper bounds of Bakry-Emery curvature.

In this note we prove convexity, in the sense of Colding-Naber, of the regular set of solutions to some complex Monge-Ampere equations with conical singularities along simple normal crossing divisors. In particular, any two points in the regular set can be joined by a smooth minimal geodesic lying entirely in the regul…

Integral estimates for Ricci flows up to singular time.

Convexity proven in Ricci shrinker limit spaces.

We define several notions of singular set for Type I Ricci flows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient shrinking solitons, thus extending work of Naber. As a by-product we conclude that the volume of a finite-volume singular s…

The study establishes inequalities on path space for sub-Riemannian manifolds.

We study blow-ups around fixed points at Type I singularities of the Ricci flow on closed manifolds using Perelman's W-functional. First, we give an alternative proof of the result obtained by Naber and Enders-Müller-Topping that blow-up limits are non-flat gradient shrinking Ricci solitons. Our second and main result …

The paper bounds the shortest closed geodesic length in 4D manifolds.

Let $\cM$ be a Brakke flow of $n$-dimensional surfaces in $R^N$. The singular set $\cS\subset\cM$ has a stratification $\cS^0\subset\cS^1\subset...\cS$, where $X\in \cS^j$ if no tangent flow at $X$ has more than $j$ symmetries. Here, we define quantitative singular strata $\cS^j_{η,r}$ satisfying $\cup_{η>0}\cap_{0<r} …

Study shows how reference measure behaves on RCD spaces through charts.

The paper examines the geometry and topology of Sasaki-Ricci solitons, proving they are either connected at infinity or compact.

Paper classifies 3D breathers and generalizes Ricci soliton results.

Improved optimal regularity for harmonic almost complex structures.

Extends positive mass theorem to arbitrary dimensions using a new inductive scheme.

The paper studies harmonic map flows and proves rectifiability of singular sets.

Study fractional Allen-Cahn equation and nonlocal minimal surfaces, improving energy and perimeter estimates.

The paper proves regularity for minimal surfaces near polyhedral cones.

New examples challenge Geroch conjecture stability.

The stability of the Yamabe invariant of $S^3$ is discussed.

Estimates on Einstein manifolds improve Brownian motion behavior and curvature limits.

Generalizes Bochner formula to path space for Ricci flow.

New examples of manifolds in tangent cones of non-collapsed Ricci limit spaces.

Gradient Ricci solitons can be extended to non-gradient Ricci solitons using energy function.

Study bounds singular set of minimal hypersurfaces with index control.

Sharp lower bounds for eigenvalues on weighted p-Laplacian manifolds.

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

Study confirms conjectures on Ricci limit spaces and their topological properties.

Paper proves ε-regularity for shrinking Ricci solitons and Ricci flows.

Researchers modify $d_p$ distance to handle long, thin splines.

This paper studies the convergence of penalized energy to harmonic maps in Riemannian manifolds.