arXiv research

We produce infinitely many examples of Anosov flows in closed 3-manifolds where the set of periodic orbits is partitioned into two infinite subsets. In one subset every closed orbit is freely homotopic to infinitely other closed orbits of the flow. In the other subset every closed orbit is freely homotopic to only one …

Constructs retractions of CAT(1) spaces to convex subsets.

The paper studies singular sets in Ricci flow limits, proving rectifiability and curvature bounds.

PixelCNN models can achieve state-of-the-art results on CIFAR-10 with exact likelihood computation.

The Kähler-Ricci flow smooths positive currents on Kähler manifolds.

B Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators $C(S)$, which are nonnegative in a suitable sense, to every $Ad_{SO(n,\C)}$ invariant subset $S \subset {\bf so}(n,\C)$. For curvature operators of a Kähler manifold of complex dimension $n$, one considers $Ad_{GL(n,\…

The space of non-singular flows on any given solenoid is shown to contain a generic subset consisting of flows that are not almost periodic. Whether this result carries over to Hamiltonian flows remains an open question.

Excises interesting subsets from symplectic manifolds.

We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows to smoothly flow through singularities by studying graphical mean curvature flow wit…

New measure defined for Brakke flow, linking classical and new definitions.

The Kähler-Ricci flow converges to a negative Kähler-Einstein metric under certain conditions.

The paper proves strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.

Let $Φ$ be a flow on a smooth, compact, finite-dimensional manifold $M$. Consider the subsets $E(Φ)$ and $D(Φ)$ of $C^{\infty}(M,M)$ consisting of smoothh mappings and diffeomorphisms (respectively) of $M$ preserving the foliation of the flow $Φ$. Let also $E_{0}(Φ)$ and $D_{0}(Φ)$ be the identity path components of $E…

Compact mean curvature flow solutions with bounded curvature in high dimensions are constructed.

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} …

Fix a $C^\infty$ principal $G$--bundle $E^0_G$ on a compact connected Riemann surface $X$, where $G$ is a connected complex reductive linear algebraic group. We consider the gradient flow of the Yang--Mills--Higgs functional on the cotangent bundle of the space of all smooth connections on $E^0_G$. We prove that this f…

In [5], Sáez and Schnürer studied the graphical mean curvature flow of complete hypersurfaces defined on subsets of Euclidean space. They obtained long time existence. Moreover, they provided a new interpretation of weak mean curvature flow. In this paper, we generalize their results to a general curvature setting. Our…

Using zippered rectangle coordinates we parametrize a Poincaré section for horocycle flow on the space of genus 2 translation surfaces with one singular cone point of angle $6π$. In addition, we bound the return time under horocycle flow to this Poincaré section by examining a subset of surfaces where a certain sum of …

Let $X$ be a compact Kähler manifold. We prove that the Kähler-Ricci flow starting from arbitrary closed positive $(1,1)$-currents is smooth outside some analytic subset. This regularity result is optimal meaning that the flow has positive Lelong numbers for short time if the initial current does. We also prove that th…

The best known finite-time local Ricci flow singularity is the neckpinch, in which a proper subset of the manifold becomes geometrically close to a portion of a shrinking cylinder. In this paper, we prove precise asymptotics for rotationally symmetric Ricci flow neckpinches. We then compare these rigorous results with …

This paper proves exponential mixing for frame flows on hyperbolic manifolds with cusps.

New convexity concept applied to sphere yields quermassintegral inequalities.

In this paper, we prove that if $M_t\subset \mathbb{R}^{n+1}$, $2\leq n\leq 6$, is the $n$-dimensional closed embedded $\mathcal{F}-$stable solution to mean curvature flow with mean curvature of $M_t$ is uniformly bounded on $[0,T)$ for $T<\infty$, then the flow can be smoothly extended over time $T$.

Study connects flow dynamics to 3D geometry via surface intersections.

Paper solves local well-posedness for Schrödinger flow into sphere with natural boundary conditions.

Geodesic flow mixing on convex projective manifolds proven.

Proves smoothness of conical singularities in mean curvature flow.

Compactifies geodesic flows on hyperbolic surfaces, revealing attractive circles at infinity.

Constructs ancient solutions to mean curvature flow with prescribed singular sets.

We show that if $Σ\subset \mathbb R^4$ is a closed, connected hypersurface with entropy $λ(Σ)\leq λ(\mathbb{S}^2\times \mathbb R)$, then the level set flow of $Σ$ never disconnects. We also obtain a sharp version of the forward clearing out lemma for non-fattening flows in $\mathbb R^4$ of low entropy.

Proves magnetic geodesic flow on sphere is integrable with constant 2-form.

We study properly immersed ancient solutions of the codimension one mean curvature flow in $n$-dimensional Euclidean space, and classify the convex hulls of the subsets of space reached by any such flow. In particular, it follows that any compact convex ancient mean curvature flow can only have a slab, a halfspace or a…

Given an embedded cylinder in an arbitrary surface, we give a gauge theoretic definition of the associated Goldman flow, which is a circle action on a dense open subset of the moduli space of equivalence classes of flat SU(2)-connections over the surface. A cylinder in a compact nonorientable surface lifts to two cylin…

New study confirms some mean curvature flow solutions have bounded mean curvature.

Paper studies convergence of Yang-Mills-Higgs flow on Kähler manifolds.

Flow of curves with curvature and forcing vector field exists.

Study on veering triangulations and their flow graphs, proving new applications.

Closed geodesics densely cover a circle in dilation surfaces.

The paper describes flows of MMD functionals with distance kernel and quantile functions.

We introduce a flow of Riemannian metrics over compact manifolds with formal limit at infinite time a shrinking Ricci soliton. We call this flow the Soliton-Ricci flow. It correspond to a Perelman's modified backward Ricci type flow with some special restriction conditions. The restriction conditions are motivated by c…

Sharp estimates for heat flow on nonconvex domains.

Compactness theory for super Ricci flows provides convergence results.

It is a theorem of S. Bando that if $g(t)$ is a solution to the Ricci flow on a compact manifold $M$, then $(M, g(t))$ is real-analytic for each $t >0$. In this note, we extend his result to smooth solutions on open domains $U\subset M$.

Study geodesic flows on cones over Riemannian manifolds, showing superintegrability.

Infinite volume found in the thick part of $\mathrm{PSL}_n(\mathbb{R})$-Hitchin-Riemann moduli space.

We prove that any complete immersed globally orientable uniformly 2-convex translating soliton $Σ\subset \mathbb{R}^{n+1}$ for the mean curvature flow is locally strictly convex. It follows that a uniformly 2-convex entire graphical translating soliton in $\mathbb{R}^{n+1},\, n\geq 3$ is the axisymmetric "bowl soliton…

Horizon saddle connections imply dense hyperbolic geodesics on dilation surfaces.

We show that for two dimensional manifolds M with negative Euler characteristic there exists subsets of the space of smooth Riemannian metrics which are invariant and either parabolic or backwards-parabolic for the 2nd order RG flow. We also show that solutions exists globally on these sets. Finally, we establish the e…