We discuss sharp estimates for singularities of
geometric flows and optimal regularity for a classical degenerate
elliptic equation.
Camillo De Lellis (Zürich, IAS):
Regularity of area minimizing currents mod \(p\).
Consider the (possibly empty) interior singular set \(S\)
of an area minimizing \(m\)-dimensional current \(T\) mod \(p\) in codimension \(n\).
In one of his first papers Brian White showed that, when \(p\) is odd and \(n=1\),
\(S\) has dimension at most \(m-1\). Prior to his work, a similar dimension bound was only
known for \(p=3\), \(m=2\) and \(n=1\) (as a consequence of a deeper result of
Taylor) and for \(p=2\) (where better bounds are in fact known since Federer's
seminal work).
Brian's paper shows that, when \(n=1\), the existence of a flat tangent cone
with multiplicity strictly less than \(p/2\) guarantees the regularity of any given
point.
In a joint work with Hirsch, Marchese and Stuvard, we prove that the singular
set \(S\) has dimension at most \(m-1\) for every \(p,m\) and \(n\). Our proof is based on
a suitable modification of Almgren's regularity theory and thus, while it
works without any restriction on the codimension, for hypersurfaces we are still
unable to rule out the existence of singular points with a flat tangent cone.
Gerhard Huisken (Tübingen, Oberwohlfach): Surgery techniques in geometric evolution equations
for hypersurfaces.
To extend solutions of geometric evolution equations such as
Ricci flow or mean curvature flow beyond singularities by surgery,
several a priori estimates have to be combined. The lecture describes
new estimates for geometric evolution equations of hypersurfaces and
their application to the classification of singularities
and the surgery procedure.
Fernando Codá Marques (Princeton): Abundance of minimal surfaces, Part II.
In recent years the existence theory of minimal hypersurfaces has enjoyed dramatic progress. We will survey the main results.
Francisco Martìn (Granada): Families of translators for mean curvature flow.
We construct new families of complete, properly embedded (non-graphical) translators: a two-parameter
family of translating annuli, examples that resemble Scherk's minimal surfaces, and examples that
resemble helicoids. This is a joint work with D. Hoffman, T. Ilmanen and B. White.
Bill Minicozzi (MIT): Arnold-Thom conjectures for the arrival time.
The arrival time for a monotone flow is twice
differentiable, but not necessarily \(C^3\). Yet, we will see that it
behaves in some ways like an analytic function. In particular, it has
properties that Thom and Arnold conjectured to hold for analytic
functions.
Aaron Naber (Northwestern): Singular set structure of spaces with lower Ricci curvature bounds.
If \(M_i \rightarrow X \) is a limit of noncollapsed manifolds which have uniform
lower bounds on Ricci curvature, then as with other geometric equations the singular
set \(Sing(X)\) of \(X\) is known to be stratified into pieces \(S^k(X)\), where \(S^k(X)\)
are the points of X for which no tangent cone is k+1 symmetric. We prove that \(S^k(X)\)
is \(k\)-rectifiable, which can be thought of as meaning \(S^k(X)\) is a \(k\)-manifold away
from a set of measure zero. We will discuss a variety of applications. The ideas involved turn
out to be distinct from those used to prove similar results in other geometric equations, for good
reason, and may be more applicable to geometric flow problems, for instance the mean curvature flow.
This is joint work with Jeff Cheeger and Wenshuai Jiang.
André Neves: Abundance of minimal surfaces, Part I .
In recent years the existence theory of minimal hypersurfaces has enjoyed dramatic
progress. We will survey the main results.
Harold Rosenberg (IMPA): Minimal planes in asymptotically flat 3-manifolds.
Laurent Mazet and I have improved a result by Chodosh and Ketover.
We prove that, in an asymptotically flat \(3\)-manifold \(M\) that contains no closed minimal surfaces,
fixing \(q\in M\) and a \(2\)-plane V in \(T_q M\), there is a properly embedded minimal plane
\(\Sigma\) in \(M\) such that \(q\in\Sigma\) and \(T_q\Sigma=V\). We also prove that fixing
three points in \(M\) there is a properly embedded minimal plane passing through these three points.
I will discuss the idea of the proof and some related questions.
Rick Schoen (Irvine): Minimal surfaces and eigenvalues
There are certain classes of minimal surfaces whose metrics arise
as extremals for eigenvalues. In this lecture we will give an overview of
how this works and discuss the current state of affairs on the extremal problem.
Tatiana Toro (Washington): Rectifiability of measures.
The question of whether a given measure is rectifiable has played an important role
in recent developments in geometric analysis and potential theory. In this talk we
will discuss some criteria that guarantee rectifiability
and will present recent work along these lines. This is partially joint work with J. Azzam
and X. Tolsa.
Neshan Wickramasekera (Cambridge):Regularity of stable codimension-1 CMC varifolds
The lecture will describe recent joint work with C.Bellettini that establishes a sharp regularity
and compactness theory for stable codimension-\(1\) CMC varifolds. The work considers codimension-\(1\)
integral \(n\)-varifolds \( V \) with generalized mean curvature locally summable to a power \(p> n\).
No useful control on the singular set can follow from this condition alone, even when \(p = \infty\), as
shown by an example due to Brakke which is singular on a set of positive \(1\)-dimensional measure.
On the other hand, the work to be described shows that if the regular parts of such a varifold
(non-empty by Allard regularity theory, but a priori potentially very small in measure) are area stationary
and stable for volume preserving deformations, and if the varifold satisfies two mild necessary
local structural conditions (only involving parts of the varifold made up of pieces that are again
regular and that come together in a regular fashion), then all is well; i.e. away from a closed set
of dimension \(\leq n-7\), the varifold consists locally of either a single smoothly embedded CMC disk
or precisely two smoothly embedded CMC disks intersecting only tangentially
and each lying on one side of the other. The work also gives an associated compactness theorem.
Curvature estimates (joint work with C. Bellettini and O. Chodosh) then follow.
