Differential Geometry Seminar Schedule for
Winter and Spring 2018
Fridays 3:00 - 3:50pm, SH 6635
1/19 Hanming Zhou, UCSB "Lens
Rigidity for a Particle in a Yang-Mills Field"
Abstract: In this talk, we consider an inverse problem related to
the motion of a classical colored spinless particle under the influence
of an external Yang-Mills potential $A$ on a compact manifold with
boundary of dimension $\geq 3$. We show that under suitable convexity
assumptions, one can recover the potential $A$, up to gauge
transformations, from the lens data of the system, namely, scattering
data plus travel times between boundary points. The talk is based on
joint work with Gabriel Paternain and Gunther Uhlmann.
1/26 Changliang Wang, McMaster University,"Perelman's functionals on compact manifolds with
isolated conical singularities"
Abstract: We extend the theory of the Perelman's functionals on compact
smooth manifolds to compact manifolds with isolated conical
singularities. For the lambda-functional, this is essentially an
eigenvalue problem for a Schrodinger operator with singular potential.
We obtain a certain asymptotic behavior of eigenfunctions near the
singularities. This asymptotic behavior plays an important role for
deriving the variation formulas of the lambda-functional and other
applications. Moreover, we show that the infimum of the W-functional
over a suitable weighted Sobolev space on compact manifolds with
isolated conical singularities is finite, and the minimizing function
exists. We also obtain a certain asymptotic behavior for the minimizing
function near the singularities. This is a joint work with Professor
Xianzhe Dai.
2/2 Sho Seto, UCSB "Learning
seminar on
needle decomposition"
2/23 Sho Seto, UCSB "Learning
seminar on
needle decomposition"
3/2 Nicholas Brubaker, California State University,
Fullerton, "A numerical method for
computing constant mean curvature surfaces with boundary"
Abstract: Constant mean curvatures (CMC) surfaces, defined as critical
points of surface area subject to a volume constraint, describe
mathematical idealizations of physical interfaces occurring between two
immiscible fluids. Accordingly, their predicted shapes give insight
into the behavior of many micro-scale systems, such as beading or
stiction in microelectromechanical system devices. However, explicitly
computing such shapes is often impossible, especially when the boundary
of the interface is fixed and/or parameters vary. In this talk, we will
propose a robust novel numerical method for computing families of
discrete versions of CMC surfaces that is based on solving a partial
differential equation (PDE) via arc-length continuation. The method
computes both stable and unstable surfaces, unlike many direct
optimization methods, and naturally identifies bifurcations. Multiple
examples will be presented to highlight the efficacy and accuracy of
the proposed approach, including the reconstruction of a branch of
asymmetric surfaces appearing from a symmetry-breaking bifurcation.
3/9 Ruobing Zhang, Stony Brook University "Nilpotent Structure and Examples of Collapsed
Einstein Spaces"
Abstract: In this talk, we will focus on the construction of new
examples of collapsed Einstein spaces. A special case is to construct a
family of hyperkähler metrics on a K3 surface which are collapsing
to a closed interval [0,1]. Geometrically, the regular collapsing
fibers in our example are 3D-Heisenberg (nilpotent) manifolds with
almost flat metrics, while the singular collapsing fibers are singular
circle fibrations over a flat torus. Moreover, there is a natural
constant mean curvature foliation from the regular fiber to the
singular fiber. Compared with the known examples of codim-1 and codim-2
collapsed Ricci-flat hyperkähler spaces, the collapsing fiber in
our example are non-abelian. We will also see how such a collapsing
phenomenon is related to a general regularity theorem.
4/6 Qiongling Li, Caltech,
"On cyclic Higgs bundles"
Abstract: Given a closed Riemann surface and a Lie group G, the
non-abelian Hodge theory gives a correspondence between the space of
representations of the surface group into G with the moduli space of
G-Higgs bundles. The correspondence is through looking for an
equivariant harmonic map to the symmetric space associated to G, to a
given representation or a given Higgs bundle. We derive a maximum
principle for a type of elliptic systems and apply it to study cyclic
Higgs bundles. We show several domination results on the pullback
metric of the (possibly branched) minimal immersion associated to
cyclic Higgs bundles. Also, we obtain a lower and upper bound of the
extrinsic curvature of the image of the minimal immersion.
4/13 Shoo Seto, UCSB, "The
first eigenvalue of the p-Laplacian on Riemann and Kahler manifolds"
4/20 Katy Craig, UCSB "Gradient
flow in the Wasserstein Metric"
Abstract: For a range of partial differential
equations–including the porous medium equation, the Fokker-Planck
equation, and the Keller-Segel equation—solutions of the equations can
be characterized as gradient flows with respect to the Wasserstein
metric on the space of probability measures. This gradient flow
structure lies at the heart of many recent analytic and numerical
results regarding questions of stability, uniqueness, and singular
limits.
Gradient flows with respect to Hilbert space norms are a classical tool
in the study of partial differential equations, but the geometry of the
Wasserstein metric presents new challenges. First, even for probability
measures on Euclidean space, the Wasserstein metric it is positively
curved in dimensions higher than one. Second, the metric lacks a
rigorous Riemannian structure, which one would normally use to make
sense of the “gradient” in a “gradient flow”. In this talk, I will
introduce a time discretization of the gradient flow problem, due to
Jordan, Kinderlehrer, and Otto, by which these problems can be overcome
and present new results which extend the convergence of the time
discrete scheme to a new class of partial differential equations of
applied interest.
5/11 Li Ma, University of Science and Technology
Beijing "Results of Lichnerowicz equations
on manifolds"
5/18 Jeffrey A Viaclovsky, UCI "Nilpotent structures and collapsing Ricci-flat
metrics on K3 surfaces"
ABSTRACT: I will discuss a new construction of families of
Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval,
with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a
corresponding singular fibration from the K3 surface to the interval,
with regular fibers diffeomorphic to either 3-tori or Heisenberg
nilmanifolds. This is joint work with Hans-Joachim Hein, Song Sun, and
Ruobing Zhang.
6/1 Luca Spolaor, MIT & Princeton "(Log-)epiperimetric inequality and regularity at
isolated singularities for almost Area-Minimizing currents"
Abstract: The uniqueness of blow-up and regularity of multiplicity-one
minimal surfaces at isolated singularities has been successfully
investigated by Allard-Almgren [Ann. of Math. '81], in the integrable
case, and by L. Simon [Ann. of Math. '83], in its full generality.
In this talk I will present a simple and completely variational
approach to this problem, achieved by proving a new logarithmic
epiperimetric inequality for multiplicity-one stationary cones with
isolated singularity. In contrast to classical epiperimetric
inequalities by Reifenberg [Ann. of Math. '64], Taylor [Invent. Math.
'73, Ann. of Math. '76] and White [Duke '83], we require no a priori
assumptions on the structure of the cone (e.g. integrability). If the
cone is integrable (not only through rotations), we recover the
classical epiperimetric inequality. Epiperimetric inequalities of
logarithmic type were first introduced by M.Colombo, B. Velichkov and
myself in the context of the obstacle and thin-obstacle problems.
As a consequence of our analysis we give a new proof of Allard-Almgren
and Simon results in the case of minimizers and we deduce a new
epsilon-regularity result for almost area-minimizing currents at
isolated singularities.
This is joint work with M. Engelstein (MIT) and B. Velichkov (Grenoble).
6/8 Xavier Ramos Olive, UCR "Li-Yau Gradient Estimate under Integral Ricci
Curvature Bounds"
Abstract: Li-Yau gradient estimates are one of the key
ingredients in many results in Geometric Analysis. To obtain them, one
usually needs to assume a lower bound on the Ricci curvature and some
convexity condition on the boundary. Following the work of Q.S.Zhang
and M.Zhu, we will describe a new Li-Yau gradient estimate under
integral Ricci curvature conditions for the Neumann heat kernel. We
will show how we can get an estimate on domains that are not
necessarily convex, but that satisfy the interior rolling $R-$ball
condition introduced by J.Wang and R.Chen.
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