Research Interests
Jesse Ratzkin
October 23, 2003
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Introduction
I am interested in problems which lie at the intersection of geometry and analysis. Typically,
these problems arise in extremizing a natural geometric quantity (e.g. length or area), leading
to a geometric property which is characterized by a dierential equation. In these problems the
geometry and analysis intertwine in complicated and beautiful ways. A common diculty in such
problems is that the geometric objects are often noncompact or develop singularities.
In the two sections below, I will describe two areas (mean curvature and scalar curvature) of
geometric analysis which currently hold great interest for me. The two main themes to my work
are constructing new examples and understanding moduli spaces of these geometric objects.
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Mean curvature of surfaces
Mean curvature is an important and active area of research with wide-ranging applications. For
instance, the energy functional which constant mean curvature surfaces extremize plays a crucial
role in AdS/CFT correspondence (see [WY], [CG] and [And]). Using methods I will describe
below, I have constructed new constant mean curvature surfaces and used this construction to
explore the moduli space of constant mean curvature surfaces. I will continue these pursuits.
Consider a surface embedded in R3 . One can deform this surface in the direction of its
normal vector. The rst variation of the area of the surface under such deformations is the mean
curvature of , commonly denoted H = H . More geometrically, the mean curvature of at a
point p is the average of the (extrinsic) curvatures of at p in all directions. The dierential
equation H = constant is a quasilinear, elliptic dierential equation on the surface . In
this section we will concentrate on surfaces with nonzero, constant mean curvature (abbreviated
CMC), which we will normalize so that H = 1. The study of CMC surfaces goes back to the
work of Gauss and Meusnier. Current research concentrates on the following two questions: How
many CMC surfaces can one construct explicitly? How well can one understand the space of all
CMC surfaces with xed topology?
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2.1 Examples of constant mean curvature surfaces
The earliest examples of CMC surfaces (with H = 1) are spheres and cylinders. In 1841, Delaunay
[Del] found all rotationally symmetric CMC surfaces, which include spheres and cylinders in a
one-parameter family. The Delaunay surfaces are periodic and characterized by the minimum
radius of a cross-section. We will call this minimum radius the necksize of the Delaunay surface.
Alexandrov [A] proved that the only compact, embedded CMC surface is a round sphere.
Meeks [M] showed that there are no one-ended CMC surfaces. Korevaar, Kusner and Solomon
[KKS] proved that the Delaunay surfaces are the only embedded, two-ended CMC surfaces.
They also proved that any end of a noncompact, embedded CMC surfaces with nite topology is
asymptotic to a Delaunay surface.
At approximately the same time as [KKS], Kapouleas [Kap] provided one of the rst gluing
constructions for CMC surfaces (see also [Sm]). In one of his constructions, Kapouleas attaches
Delaunay ends with very small necksizes onto spheres, constructing families of surfaces with nearly
constant mean curvature and perturbing them to have constant mean curvature. The analysis
required for this construction is quite delicate, because the perturbation is nonlinear and global
in nature. Later, Mazzeo and Pacard [MP] attached Delaunay ends onto a minimal surface core,
using a dierent technique. Instead of building surfaces with nearly constant mean curvature and
perturbing them, Mazzeo and Pacard solve a general boundary value problem to nd many CMC
surfaces with boundary. They then match the Cauchy data of the solutions across an interface.
Using this technique, Mazzeo, Pacard and Pollack have constructed connected sums of CMC
surfaces [MPP1] and attached a Delaunay end onto a CMC surface [MPP2].
I attached CMC surfaces together in an end to end fashion [R1], provided the ends are asymptotic to the same Delaunay surface. In this construction, one truncates the ends of the two summands and glues along these two boundary components. One can use either the perturbation
method or the method of matching Cauchy data to perform the end to end gluing. In [MPPR] we
construct CMC surfaces of arbitrary genus and three or more ends, along with other interesting
CMC surfaces.
2.2 Moduli space theory: the totality of all constant mean curvature
surfaces
One hopes these gluing constructions will provide us with a better understanding of the moduli
space of CMC surfaces. We dene the moduli space
MCMC := fembedded CMC surfaces of genus g having k endsg:
k;g
We endow MCMC with the Hausdor topology on compact sets. In [KMP], Kusner, Mazzeo
and Pollack showed that for k 3, MCMC is a real-analytic variety of formal dimension 3k.
They also showed that if the linearization L of the mean curvature operator is injective on
L2 () then MCMC is a real-analytic manifold in a neighborhood of . One can compute that
L = + jA j2 , where is the Laplacian of the induced metric on and A is the second
fundamental form. We call a surface nondegenerate if L is injective on L2 (). This is
equivalent to the condition that zero is not in the closure of the L2-spectrum of L .
k;g
k;g
k;g
Nondegeneracy is a very useful property. Indeed, many gluing constructions assume that
the summands are nondegenerate. Nondegenerate CMC surfaces exist; Delaunay surfaces are
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nondegenerate, as are all the surfaces constructed in [MP], [MPP1], [MPP2] and [R1]. However,
it is dicult to tell which surfaces are nondegenerate. I am working with several people, primarily
Nick Korevaar and Rob Kusner, to prove that genus zero CMC surfaces with three ends are
nondegenerate. The topological classication result of Grosse-Brauckmann, Kusner and Sullivan
[GKS] strongly hints that all these surfaces are nondegenerate. We have some preliminary results
and a better understanding of genus zero CMC surfaces with three ends, although we have not
yet proven nondegeneracy.
One can use gluing constructions to determine the topology of MCMC . The end to end gluing
construction of [R1] shows that MCMC is not simply connected for k 4. Along the same lines,
[MPP2] gives lower bounds for the complexity of 1 (MCMC ) using another gluing construction.
Various gluing constructions and moduli space results indicate that, at least in the genus zero
case, MCMC
0 inherits much of its topological structure from a natural object in Teichmuller theory.
k;g
k;g
k;g
k;
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Scalar curvature
Geometrically, scalar curvature measures the volume growth of small geodesic spheres of a Riemannian manifold (M; g). The study of constant scalar curvature metrics relates topology, geometry and analysis in many ways. For instance, there are topological restrictions on which compact
manifolds admit metrics with constant positive scalar curvature. (There is a large body of work
in this area, by Gromov, Lawson, Stoltz, Botvinnik, J. Rosenberg and others; see [GL], [BR] and
[St].) Also, one can nd the best constant for the Sobolev inequality in R by examining constant
scalar curvature metrics on S . The most famous result regarding scalar curvature the solution
to the Yamabe problem; see section 3.1.
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I have constructed new, complete metrics of constant, positive scalar curvature on punctured
spheres [R2], using a method which is similar to the construction of [R1]. I will pursue further
constructions and moduli space results in this area, as well as the mysterious parallel between
mean curvature and scalar curvature.
3.1 The Yamabe and singular Yamabe problems
Given a Riemannian manifold
(M; g) of dimension n > 2, one might wish to deform the metric by a
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conformal factor to g~ = u n?2 g such that g~ is more uniform. Under this conformal transformation,
~ nn?+22 = 0;
u ? 4(nn??21) Ru + 4(nn??21) Ru
(1)
where R and R~ are the scalar curvatures of g and g~, respectively. This is a semilinear, elliptic
equation in u, where the nonlinearity has critical Sobolev growth.
g
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The Yamabe problem asks for a conformal metric g~ = u n?2 g with constant scalar curvature.
This is equivalent to nding a positive solution u to equation (1) with R~ constant. By a celebrated
theorem of Yamabe, Aubin and Schoen (see [Au], [S1] and [LP]), the Yamabe problem always
has a solution when M is compact.
In solving the Yamabe problem for compact manifolds, one is naturally led to the singular
Yamabe problem: given a compact Riemannian manifold (M; g) and a closed set M satisfying
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some regularity condition, nd a conformally related metric g~ = u n?2 u with constant scalar
curvature which is complete on M n. That g~ is complete on M n forces u to blow up along .
The type of solution one can nd is related to the dimension of the singular set . Aviles and
McOwen [AM] showed that if is a closed, smooth submanifold then there is a singular Yamabe
metric with constant, negative scalar curvature if and only if the dimension of is greater than
(n ? 2)=2. The models to consider in this case are M = S and = S . Then M n = S nS is
conformally equivalent to S ? ?1 H +1 , where H +1 is hyperbolic space. This product manifold
has scalar curvature (n ? 2)(n ? 1) ? 2k(n ? 1), which is negative precisely when k > (n ? 2)=2.
Schoen and Yau [SY] prove similar theorems in the positive scalar curvature case.
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k
k
k
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k
k
3.2 The case of a nitely punctured sphere
The case when (M; g) is a round sphere and a nite number of points is one of the more
complicated instances of the singular Yamabe problem. As in the case of noncompact CMC
surfaces with nite topology, Caarelli, Gidas and Spruck [CGS] show that the metric must
asymptotically approach a rotationally symmetric metric as one approaches a puncture point.
These rotationally symmetric metrics behave much like the Delaunay surfaces, and are called
Delaunay metrics.
As in the case of CMC surfaces, there is a moduli space theory for these singular Yamabe
metrics [MPU]. There are also gluing constructions of singular Yamabe metrics for a punctured
sphere, starting with Schoen's construction [S2]. In particular, I showed that one can perform
end to end gluing in for singular Yamabe metrics on punctured sphere [R2].
A conjecture of Schoen states that if a singular Yamabe metric on a nitely punctured sphere
is asymptotically cylindrical at each end then it is a cylinder. A related project I am working on
with Rob Kusner is to nd a priori necksize bounds for singular Yamabe metrics, using a theorem
of Pollack [P] (see also [S2]) which relates the necksize to an integral over a hypersurface. This
integral is a homology invariant of the hypersurface. If the metric has symmetry, there are several
natural choices for the hypersurface which take advantage of the symmetry, and could yield a
nontrivial necksize bound. This work in progress could prove a special case of Schoen's conjecture.
Also, as was remarked in [KK] and [MPo], there is a mysterious parallel between noncompact,
embedded CMC surfaces with nite topology and singular Yamabe metrics on punctured spheres.
Most of the analysis for one problem will carry over to the other (e.g., compare [KMP] and
[MPU]), even though one is a problem in extrinsic geometry and the other is a problem in
intrinsic geometry. No one has completely explained this parallel.
Can one possibly push this parallel farther? Korevaar and Kusner [KK] proved a global
structure theorem for CMC surfaces, showing that any such surface in contained in a union of
solid cylinders of radius 6 and solid balls of radius 21. They also give bounds on the number of
these cylinders and balls, depending on the topology of the surface. However, there is currently
no parallel theorem in the realm of singular Yamabe metrics. One of the problems with producing
such a theorem is that it is not clear what the right conguration space should be. Also, it is not
clear how to nd sharp bounds for the conformal factor u away from the singular set. (There are
sharp bounds for u near the singular set; see [P] and [KMPS]. These bounds near the singular set
prove a \cylindrical boundedness" result similar to the result in [M].) Finally, there is currently
no scalar curvature parallel for the construction of CMC surfaces of Grosse-Brauckmann [G],
which was used in [GKS]. All of these are interesting and worthwhile questions I will explore.
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References
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[MPP2] R. Mazzeo, F. Pacard and D. Pollack. The conformal theory of Alexandrov embedded
constant mean curvature surfaces in R3 . to appear: Proceedings of the 2001 Clay/MSRI
workshop on the global theory of minimal surfaces.
[MPPR] R. Mazzeo, F. Pacard, D. Pollack and J. Ratzkin. Constant mean curvature surfaces in
Euclidean three-space via gluing constructions. in preparation.
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[R1] J. Ratzkin. An end to end gluing construction for surfaces of constant mean curvature. Ph.D.
Thesis, University of Washington, 2001.
[R2] J. Ratzkin. An end to end gluing construction for metrics of constant positive scalar curvature. Indiana Univ. Math. J. 52:703{726, 2003.
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