Schedule for: 21w5504 - New Directions in Geometric Flows

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In 1994 Velázquez constructed a smooth \( O(4)\times O(4)\) invariant Mean Curvature Flow that forms a type-II singularity at the origin in space-time. Stolarski very recently showed that the mean curvature on this solution is uniformly bounded. Earlier, Velázquez also provided formal asymptotic expansions for a possible smooth continuation of the solution after the singularity. Jointly with S. Angenent and N. Sesum we establish the short time existence of Velázquez' formal continuation, and we verify that the mean curvature is also uniformly bounded on the continuation.

We propose a class of local definitions of weak lower scalar curvature bounds that is well defined for C^0 metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.

Turn on the cameras in zoom for virtual photo. In-person participants, meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension n ≥ 4, we find a family of Z2 × O(n − 1)-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operators. We show that these solitons are non-collapsed.

Mean curvature flow can be continued through singularities via Brakke flow or level set flow. Brakke flow is defined with an inequality which makes it tantamount to a subsolution to smooth mean curvature flow. On the other hand, level set flow is like a supersolution, since it may attain positive measure. In this talk, we will discuss these weak solutions and will relate uniqueness problems for weak solutions to multiplicity problems in mean curvature flow. In particular, we discuss how Brakke flows with only generic singularities achieve equality in the inequality defining the Brakke flow. This uses an analysis of worldlines in the Brakke flow, analogous to the theory of singular Ricci flows.

In this talk, I will discuss the relationship between Inverse Mean Curvature Flow (IMCF), an expanding extrinsic geometric flow, and minimal surfaces. A natural question about the IMCF of a closed hypersurface in Euclidean space is whether a finite-time singularity forms. When one does form, I will show how classical minimal surfaces may be used to characterize the flow behavior near the singular time: specifically, they allow one to establish a uniform bound on total curvature and hence a limit surface without rescaling the flow surfaces at the extinction. This singular profile contrasts sharply with the singular profiles of other extrinsic flows. When one does not form and the evolution continues for all time, there is a connection to previous work by Meeks and Yau on the embedded Plateau problem. In particular, via a comparison principle arising from embedded global solutions of IMCF, I will show that global area-minimizers for Jordan curves confined to star-shaped or certain rotationally symmetric mean-convex surfaces in $R^3$ are embedded. Furthermore, such curves admit only a finite number of stable minimal disks with areas smaller than any fixed number.

The Kahler-Ricci flow admits a long-time solution if and only if the canonical bundle of the underlying Kahler manifold is nef. We prove that if the canonical bundle is semi-ample, the diameter is uniformly bounded for long-time solutions of the normalized Kahler-Ricci flow. Our diameter estimate combined with the scalar curvature estimate for long-time solutions of the Kahler-Ricci flow are natural extensions of Perelman's diameter and scalar curvature estimates for short-time solutions on Fano manifolds.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will describe recent work with Choi, Mantoulidis, Schulze concerning generic behavior of MCF. I will compare two potential approaches to this problem and describe one of them (based on entropy drop near non-generic singularities) in detail.

Translating solution to the mean curvature flow form, together with self-shrinking solutions, the most important class of singularity models of the flow. When a translator arises as a blow-up of a mean convex mean curvature flow, it also naturally satisfies a noncollapsing condition. In this talk, I will report on a recent work with Kyeongsu Choi and Robert Haslhofer, in which we show that every mean convex, noncollapsed, translator in $R^4$ is a member of a one parameter family of translators, which was earlier constructed by Hoffman, Ilmanen, Martin and White.

Łojasiewicz inequalities are a popular tool for studying the stability of geometric structures. For mean curvature flow, Schulze used Simon’s reduction to the classical Łojasiewicz inequality to study compact tangent flows. For round cylinders, Colding and Minicozzi instead used a direct method to prove Łojasiewicz inequalities. We’ll discuss similarly explicit Łojasiewicz inequalities and applications for other shrinking cylinders and products of spheres.

In 1984, Huisken showed that the second fundamental form always blows up at a finite-time singularity for the mean curvature flow. Naturally, one might then ask if the mean curvature must also blow up at a finite-time singularity. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular with uniformly bounded mean curvature.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The talk will survey Ricci flow through singularities in dimension three, and some applications to topology; the lecture is intended for nonexperts. This is joint work with Richard Bamler and John Lott.

In this talk I will overview joint work with J. Jordan and J. Streets, in arXiv:2106.13716, about Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form. These metrics give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow, as introduced by Streets and Tian, implying new global existence results. In particular, on all complex non-Kähler surfaces of nonnegative Kodaira dimension. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We prove an optimal relative integral convergence rate for two expanding gradient Ricci solitons coming out of the same cone. As a consequence, we obtain a unique continuation result at infinity and we prove that a relative entropy for two such self-similar solutions to the Ricci flow is well-defined. This is joint work with Alix Deruelle.

In both Ricci flow and mean curvature flow, there have recently been significant advances in our understanding of ancient solutions which model singularity formation. One of the crucial tools to this advance has been the development of local symmetry improvement results, as first introduced in mean curvature flow by Brendle and Choi, and later to the Ricci flow by Brendle. In this talk, we would like to discuss how the technique can be adapted to higher codimension mean curvature flow, exhibiting how both rotational symmetry and flatness improve along the flow.

In this joint work with B. Zhu, we prove that for every three dimensional manifold with non-negative Ricci curvature and strictly mean convex boundary, there exists a Morse function so that each connected component of its level sets has a uniform diameter bound, which depends only on the lower bound of mean curvature. This gives an upper bound of Uryson 1-width for those three manifolds with boundary. Our proof uses mean curvature flow with free boundary proved by Edelen-Haslhofer-Ivaki-Zhu.

We show that any closed connected hypersurface with sufficient low entropy is smoothly isotopic to the standard round sphere.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We will discuss recent results and progress made on classifying ancient solutions in geometric flows. We will also mention very nice applications of these results that play an important role in singularity analysis of mean curvature flow and Ricci flow.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.

Steady gradient Kähler-Ricci solitons are fixed points of the Kähler-Ricci flow evolving only by the action of biholomorphisms generated by a real holomorphic vector field. We show that there is a unique steady gradient Kähler-Ricci soliton in each Kähler class of a crepant resolution of a Calabi-Yau cone. To do this, we solve a complex Monge-Ampere equation via a continuity method. Our construction is based on an ansatz due to Cao in the 90’s which was utilized by Biquard-MacBeth in 2017. This is joint work with Alix Deruelle.