Jacob Bernstein (Johns Hopkins)
Title: The Asymptotic Geometry of Genus-g Helicoids
Abstract: We (briefly) discuss the problem of classifying the asymptotic geometry of complete, properly embedded minimal surfaces in $R^3$ with finite topology. We will focus on the case of surfaces with one end and sketch the tools needed to address this question -- namely the theory of Colding and Minicozzi.
Cristina Caputo (U. Texas, Austin)
Title: Non-local quasiminimal surfaces
Abstract: We introduce a notion of non-local quasiminimal surfaces similar to that appearing in geometric measure theory for the perimeter. Using recent methods developed for non-local minimal surfaces we prove that flat, non-local quasiminimal surfaces are smooth, which can be viewed as a non-local version of the Almgren-DeGiorgi-Tamanini regularity theory for quasiminimal bound- aries. The main result has several applications, among these C 1,alpha regularity for sets with prescribed non-local mean curvature in Lp.
Maria Calle (U. Autonoma de Madrid)
Title: Width and flow of hypersurfaces by curvature functions
Abstract: Sweepouts of a closed manifold by closed curves have been used to find closed geodesics on a given homotopy class. Colding and Minicozzi constructed a sweepout such that if a curve in the sweepout has energy close to the width, then the curve is close to a geodesic. They used this result to give a bound on the extinction time for mean curvature flow. We generalize this bound to a broader class of geometric flows that behaves similarly to mean curvature flow when evolving convex hypersurfaces.
This is joint work with Stephen Kleene and Joel Kramer.
Julie Clutterbuck (ANU, Canberra)
Title: Estimate for the fundamental gap
Abstract: We consider the eigenvalues of the Schrodinger operator $\Delta-V$. This operator arises in the study of minimal surfaces, in which case the potential $V$ is a curvature term, and the existence of a positive first eigenvalue is a requirement for the existence of stable minimal surfaces.
The fundamental gap is the difference between the first two eigenvalues. It is conjectured (by Yau and van den Berg) that for convex planar domains and convex potentials, the gap is minimized by a long, thin domain with a constant potential. We prove this conjecture, and also characterize the gap for a larger class of potentials.
This is joint work with Ben Andrews.
Peter Connor (Indiana University)
Title: The construction of doubly periodic minimal surfaces via balance equations
Abstract: Using Traizet's regeneration method, we prove the existence of many new 3-dimensional families of embedded doubly periodic minimal surfaces. All these families have a foliation of 3-dimensional Euclidean space by vertical planes as a limit. In the quotient, these limits can be realized conformally as noded Riemann surfaces, whose components are copies of the complex plane with finitely many nodes. We derive the balance equations for the location of the nodes and exhibit solutions that allow for surfaces of arbitrarily large genus and number of ends in the quotient.
Camillo De Lellis (Zurich)
Title: Min--max constructions of minimal surfaces
Abstract: If a Riemannian $2$-dimensional manifold $M$ is not simply connected, producing a nontrivial closed geodesic is relatively simple: it suffices to minimize the length among all noncontractible loops. Obviously, this method does not apply when $M$ is diffeomorphic to the sphere. Nonetheless, in a pioneering work which appeared in 1917, Birkhoff showed the existence of at least a nontrivial closed geodesic even in the latter case.
The proof of Birkhoff is probably the first ``min-max'' argument in the calculus of variations. In this talk we will address higher dimensional versions of his method, showing how to produce minimal hypersurfaces. In particular, we will present a new simpler approach to the existence of nontrivial embedded minimal hypersurfaces in closed Riemannian manifolds of dimension $n+1$. This theorem was first proved in a monograph of Pitts for $n\leq 5$ and extended by Schoen and Simon to the general case.
Jasun Gong (Pittsburgh)
Title: Regularity of Quasi-minimizers for Non-homogeneous Energy
Functionals on Metric Spaces
Abstract: It is well-known that on domains on Euclidean spaces,
minimizers of the p-Dirichlet energy integral enjoy a rich regularity
theory. For instance, they are Holder continuous and satisfy the
Harnack inequality. As shown by Giaquinta and Giusti in the 1980s,
"quasi-minimizers" (roughly speaking, functions which almost minimize
energy) also have similar regularity properties.
Many notions of analysis, such as Sobolev spaces, extend to the
setting of metric measure spaces. In this setting, we will show that
quasi-minimizers of energy integrals -- both homogeneous and
non-homogeneous -- have similar regularity properties as their
Euclidean counterparts. These results extend the work of J. Kinnunen
and N. Shanmugalingam, as well as of J. Bjorn and N. Marola. This is
based on joint work with J.J. Manfredi and M. Parviainen.
Zheng Huang (CUNY, Staten Island)
Title: Mean curvature flows in quasi-Fuchsian manifolds
Abstract: Quasi-Fuchsian three manifold is an important class of hyperbolic three spaces. The space of such three manifolds is a complex manifold of dimension 6g-6, where g is the genus of any incompressible closed surface in a quasi-Fuchsian three manifold. In a subclass of the same dimension, we deform an rather arbitrary closed incompressible surface, which is a graph over a fixed reference surface of small principal curvatures, to a nearby embedded minimal surface. The deformation is made transparent by the mean curvature flow.
Sevvandi Kandanaarachchi (Monash University, Melbourne)
Title : Volume preserving axially symmetric mean curvature flow
Abstract: We look at an axially symmetric hypersurface between two parallel planes, having orthogonal Neumann boundary data, evolving by mean curvature, while preserving its enclosed volume. We employ a rescaling technique to gain insight about the singularities of the flow.
Stephen Kleene (Johns Hopkins)
Title: Rotational Self Shrinkers
Abstract: (joint work with N. M. Moller, MIT). We construct a family of rotationally symmetric surfaces with boundary that are asymptotic to cones, with potential applications to finding high genus self shrinkers. We also mention, in passing, a short proof that the round sphere is the only rotational genus zero self-shrinker
Nam Le (Columbia)
Title: Blow up of sub-critical quantities at the first singular time of the mean curvature flow
Niko Marola (Helsinki)
Title: On a theorem of Beckenbach and Rado for subharmonic functions
Abstract: Beckenbach and Rado characterized logarithmically subharmonic functions in the plane in terms of integral inequalities involving spherical averages. We extend this result to higher dimensions and thus answer a question raised by Beckenbach and Rado. We also consider related integral inequalities suggested by Beckenbach and Rado and discuss connections to Muckenhoupt weights.
Andre Neves (Imperial College)
Title: Stability of singularities for LMCF
Abstract: I will detail an approach to construct compact, embedded, and zero-Maslov class Lagrangians that develop finite time singularities under the flow.
Marlio Paredes (U. of Turabo, Puerto Rico)
Title: The Combinatorial geometry of classical flag manifolds
Abstract: In this work we present some results on the geometry of the classical flag manifolds F(n). Mainly we discuss the relation between a class of directed graphs, almost complex structures and metrics on flag manifolds. The results presented here have been obtained joint with Sofia Pinzon, Caio J. Negreiros, Nir Cohen, Luis A. B. San Martin and Brendan McKay.
Felix Schulze (Freie U. Berlin)
Title: New stability results for Mean Curvature flow and Ricci flow
Abstract: For entire graphs evolving under mean curvature flow, we show that if initially the second fundamental form decays appropriately near spatial infinity then the corresponding solution is stable under graphical mean curvature flow.
For Ricci flow we consider metrics, which are C^0-perturbations of the hyperbolic metric on H^n. If the perturbation is bounded in L^2 and small enough in C^0, we show that for dimensions 4 and higher the scaled Ricci harmonic map heat flow of such a metric converges exponentially fast in C^\infty to the hyperbolic metric as time approaches infinity. We also prove a related result for Ricci flow and for the 2-dimensional conformal Ricci flow.
This is joint work with J. Clutterbuck / O. Schnuerer and O. Schnuerer / M. Simon.
Natasa Sesum (Penn)
Title: The extension results for solutions of the Ricci flow and the mean curvature flow
Abstract: We will discuss some sufficient conditions which ensure having a smooth solution to the Ricci flow and the mean curvature flow. This can be seen as sharpening the results of Hamilton that the norm of the Riemannian curvature blows up at a finite singular time and of Huisken that the norm of the second fundamental form blows up at a finite singular time.
Oliver Thomys (Leipzig)
Title: Asymptotic behavior of capillary problems governed by disjoining pressure potentials
Ray Treinen (Kansas State)
Title: On the symmetry of solutions to some floating drop problems.
Abstract: We study configurations of three fluids where one of the three fluids
has significantly less volume than the other two, and is called the
drop. If the density of the drop is between that of the other two
fluids, then it is a light drop; if the density of the drop is more
than that of both the other two fluids, then it is a heavy drop. The
main result presented is a theorem that shows that some floating drops
must be symmetric about a vertical line. This result applies to both
light and heavy drops.
Mu-Tao Wang (Columbia)
Title: Mean curvature flows and isotopy problem
Abstract: I shall discuss how mean curvature flows give canonical (symmetric) deformation of maps between Riemannian manifolds. Applications include estimations of null-homotopy constants of maps between spheres and retractions of the symplectomorphism groups of Riemann surfaces and complex projective space
Matthias Weber (Indiana University)
Title: Balance Equations for Minimal Surfaces
Abstract: In the past few years, the construction of minimal surfaces in Euclidean space has been dominated by Traizet's regeneration technique that offers a machinery to obtain minimal surface families from solutions of a set of algebraic equations. I will explain this machinery in a very simple example that avoids all technicalities, and give an overview of recent developments
Michael Wolf (Rice)
Title: Teichmuller theory and minimal surfaces in space
Abstract: We describe some applications of Teichmuller theory to problems in classical minimal surface theory. The bulk of the talk will describe work with Weber viewed through the prism of a result on the existence of doubly periodic minimal surfaces in space of increasing genus, with some additional remarks on work of Douglas on families obtained by shearing these examples.