arXiv research

Constructs a mean curvature flow with surgery for compact mean convex hypersurfaces.

Generalizes surgery techniques for projectively Anosov flows.

New surgery method preserves Anosov flow properties using bi-contact geometry.

Paper shows Ricci flow with surgery converging to Taub-NUT metric.

Smooth flows with surgery approximate weak mean curvature flows with spherical and neck-pinch singularities.

We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. We consider the behavior of Ricci flow with surgery starting from a fixed initial compact Riemannian 3-manifold, as the surgery parameter varies. We prove that the flow with surgery subconverges to a singular Ri…

Horizontal surgery on pseudo-Anosov flows yields almost equivalent flows.

Proves existence of mean curvature flow with surgery for free boundary surfaces.

New method for geometric flows with surgery without smooth estimates.

We prove a monotonicity formula for mean curvature flow with surgery. This formula differs from Huisken's monotonicity formula by an extra term involving the mean curvature. As a consequence, we show that a surgically modified flow which is sufficiently close to a smooth flow in the sense of geometric measure theory is…

Bi-contact surgery operations can be applied to Anosov flows.

Huisken and Sinestrari have recently defined a surgery process for mean curvature flow when the initial data is a two-convex hypersurface. The process depends on a parameter H. Its role is to initiate a surgery when the maximum of the mean curvature of the evolving hypersurface becomes H, and to control the scale at wh…

In this article, we extend the mean curvature flow with surgery to mean convex hypersurfaces with entropy less than $Λ_{n-2}$. In particular, 2-convexity is not assumed. Next we show the surgery flow with just the initial convexity assumption $H - \frac{\langle x, ν\rangle}{2} > 0$ is possible and as an application we …

Rotationally invariant Ricci flows are constructed and shown to converge to spacetimes.

New method for high-dimensional submanifolds using surgery and curvature control.

New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.

Shows Anosov flows with genus one sections, supporting a conjecture.

Classifies Anosov flows on figure-eight knot surgeries.

Paper shows how to transform certain flows into R-covered ones.

Computing uniformization maps for surfaces has been a challenging problem and has many practical applications. In this paper, we provide a theoretically rigorous algorithm to compute such maps via combinatorial Calabi flow for vertex scaling of polyhedral metrics on surfaces, which is an analogue of the combinatorial Y…

The paper establishes a discrete uniformization theorem for surfaces with piecewise hyperbolic metrics.

We prove a uniform Sobolev inequality for Ricci flow, which is independent of the number of surgeries. As an application, under less assumptions, a non-collapsing result stronger than Perelman's $κ$ non-collapsing with surgery is derived. The proof is shorter and seems more accessible. The result also improves some ear…

In this article we use the mean curvature flow with surgery to derive regularity estimates for the level set flow going past Brakke regularity in certain special conditions allowing for 2-convex regions of high density. We also show a stability result for the plane under the level set flow.

The paper proves existence and behavior of Lagrangian tori in complex projective plane.

In this paper, we introduce a parameterized discrete curvature ($α$-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is established for the parameterized discrete curvature, which generalizes the discrete uni…

In this paper, we construct smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches on S^(n+1), without performing an intervening surgery. In the restrictive context of rotational symmetry, this construction gives evidence in favor of Perelman's hope for a "can…

The study finds infinitely many periodic orbits that can be used to modify Anosov flows.

Legendrian arcs connect veering triangulations to Anosov flows.

Alexandrov immersed surfaces maintain their properties under mean curvature flow.

We give a new proof for the existence of mean curvature flow with surgery of 2-convex hypersurfaces in $R^N$, as announced in arXiv:1304.0926. Our proof works for all $N \geq 3$, including mean convex surfaces in $R^3$. We also derive a priori estimates for a more general class of flows in a local and flexible setting.

In the following series of papers we analyze the long-time behavior of 3 dimensional Ricci flows with surgery. Our main result will be that if the surgeries are performed correctly, then only finitely many surgeries occur and after some time the curvature is bounded by $C t^{-1}$. This result confirms a conjecture of P…

In this article, we use the recently developed mean curvature flow with surgery for 2 convex hypersurfaces to prove several isotopy existence and finally extrinsic finiteness results (in the spirit of Cheeger's compactness theorem) for the space of 2 convex hypersurfaces in $\mathbb{R}^{n+1}$.

The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.

In \cite{Luo0}, Feng Luo conjectured that the discrete Yamabe flow will converge to the constant curvature PL-metric after finite number of surgeries on the triangulation. In this paper, we prove that the flow can always be extended (without surgeries) to a solution that converges exponentially fast to the constant cur…

We decribe and announce some results (joint with G. Besson, L. Bessieres, M. Boileau and J.Porti) about the geometry and topology of 3-manifolds. Most of the article is primarily intended as an introduction for nonexperts to geometrization of 3-manifolds, Ricci flow with surgery, and the simplicial volume approach to c…

In this second part of a series of papers on the long-time behavior of Ricci flows with surgery, we establish a bound on the evolution of the infimal area of simplicial complexes inside a 3-manifold under the Ricci flow. This estimate generalizes an area estimate of Hamilton, which we will recall in the first part of t…

This is the fourth and last part of a series of papers on the long-time behavior of 3 dimensional Ricci flows with surgery. In this paper, we prove our main two results. The first result states that if the surgeries are performed correctly, then the flow becomes non-singular eventually and the curvature is bounded by $…

We define a notion of mean curvature flow with surgery for two-dimensional surfaces in $\mathbb{R}^3$ with positive mean curvature. Our construction relies on the earlier work of Huisken and Sinestrari in the higher dimensional case. One of the main ingredients in the proof is a new estimate for the inscribed radius es…

Study nondegenerate singularities in mean curvature flow.

Recently Brendle-Huisken introduced a fully nonlinear flow $G$. Their aim was to extend the surgery algorithm of Huisken-Sinestrari, into the Riemannian setting. The aim of this paper is to go through the details on how to perform neck detection for a closed, embedded hypersurface $M_0$ in $\mathbb{R}^{n+1}$ undergoing…

Fractional combinatorial flow improves surface conformal structures.

These are detailed notes on Perelman's papers "The entropy formula for the Ricci flow and its geometric applications" and "Ricci flow with surgery on three-manifolds".

We study the formation of finite time singularities of the Kahler-Ricci flow in relation to high codimensional birational surgery in algebraic geometry. We show that the Kahler-Ricci flow on an n-dimensionl Kahler manifold contracts a complex submanifold $\mathbb{P}^m$ with normal bundle $\oplus_{j=1}^{n-m}\mathcal{O}_…

In this paper we analyze the long-time behavior of 3 dimensional Ricci flows with surgery. Our main result is that if the surgeries are performed correctly, then only finitely many surgeries occur and after some time the curvature is bounded by $C t^{-1}$. This result confirms a conjecture of Perelman. In the course of…

The paper studies hypersurfaces in spheres using mean curvature flow with surgery.

In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds. In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics. In the case of surfaces, we define the combinatorial Yamabe flow on the space of all piecewise flat metrics associated to a triangul…

This is the first of a series of papers on the long-time behavior of 3 dimensional Ricci flows with surgery. In this paper we first fix a notion of Ricci flows with surgery, which will be used in this and the following three papers. Then we review Perelman's long-time estimates and generalize them to the case in which …

This is a technical paper, which is a continuation of math.DG/0211159. Here we construct Ricci flow with surgeries and verify most of the assertions, made in section 13 of that e-print; the exceptions are (1) the statement that manifolds that can collapse with local lower bound on sectional curvature are graph manifold…