Bay Area Differential Geometry Seminar (BADGS)
BADGS May 10 2014
University of California, Santa Cruz
Professor Arthur E. Fischer
University of California Santa Cruz
Title: Conformal Ricci Flow and Conformal Reduction of Einstein’s Evolution Equations of
We introduce a variation of the classical Ricci flow equation that modifies the unit volume
constraint of that equation to a scalar curvature constraint. The resulting equations are named the conformal Ricci flow equations because of the role that conformal geometry plays in constraining the scalar
curvature. These new equations are given by
+ 2 Ric(g)+ n1 g = −pg
R(g) = −1
for a dynamically evolving metric g and a non-dynamical scalar field p, known as the conformal pressure.
The conformal Ricci flow equations are analogous to the Navier-Stokes equations of fluid mechanics,
+ ∇v v + ν∆v = −grad p
div v = 0.
Just as the real physical pressure in fluid mechanics serves to maintain the incompressibility constraint of the
fluid (div v = 0), the conformal pressure serves as a Lagrange multiplier to conformally deform the metric
flow so as to maintain the scalar curvature constraint (R(g) = −1).
The conformal Ricci flow equations can be thought of as a Navier-Stokes equation for the metric, just
as the Ricci flow equation can be thought of as a heat equation for the metric. A variety of properties of
the conformal Ricci flow equations are discussed, as well as their interpretation as a flow on the Teichmüller
space of conformal structures on a compact manifold. In addition, we note that the Ricci flow equations
can be interpreted as a parabolic model for the Einstein evolution equations of general relativity, which is a
local autonomous Hamiltonian system with global constraints. Similarly, the conformal Ricci flow equations
can be interpreted as a parabolic model for the conformally reduced Einstein evolution equations of general
relativity, which is an unconstrained non-local time-dependent Hamiltonian system, as the constraints have
already been implicitly solved and removed during the process of reduction. Thus the remaining variables of
the system are freely specifiable dynamical variables of the unconstrained system. Thus conformal reduction
of Einstein’s equations gives one solution to the classical problem of writing the Einstein evolution equations
as an unconstrained Hamiltonian system.
Lastly, we remark that in conformal Ricci flow, the volume is a strictly monotonically decreasing function
of time away from equilibrium points. In conformally reduced Einstein’s equations, the reduced Hamiltonian
behaves similarly, being a strictly monotonically decreasing function of time away from equilibrium points. In
both cases, we relate this monotonicity to various topological invariants of the underlying manifold M and in
the conformal Ricci flow, we propose to use this naturally occurring strictly monotonically decreasing volume
as a Perelman type entropy to move the underlying manifold M toward geometrization under conformal Ricci
Professor Gerhard Huisken
Mathematisches Forschungsinstitut Oberwolfach and Tübingen University
Professor Ivan Sterling
St Mary’s College of Maryland
Title: Minding’s Theorem for Low Degrees of Differentiability
Abstract: Let D be an simply connected open domain in R2 and f,g : D :−→ R3 immersions of class C n .
For n ≥ 3 Gauss’s Theorem says that if the induced metrics for f and g agree then their intrinsic curvatures
agree, while Minding’s Theorem says that if their intrinsic curvatures agree and are constant then their
induced metrics agree. What happens for n = 2? For a C 2 immersion, the extrinsic curvature (determinant
of the differential of the Gauss map) is defined, but the induced metric is only C 1 , so the intrinsic curvature
may not be defined. Our main result is: If f is C 2 with extrinsic curvature K ≡ −1, then there exists a C 2
isometry φ : (D, f ∗ ds2R3 ) −→ (H2 , ds2H2 ), where f ∗ ds2R3 denotes the induced metric on f and ds2H2 denotes the
standard metric on H2 . In addition to the distributional proof some motivation and history will be included.
Professor Paul Yang
Title: CR Structures in 3-D
Abstract: I will report on on-going work to study the global CR invariant described by the Q’ curvature
integral on a 3-D CR manifold.
Hahn to McHenry
and Pedestrian Map
Drive to Hahn Services Lot and Park Science &
to UCSC. If you are visiting
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After parking, walk to McHenryGLibrary
via the route indicatedClaby
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t Wa to McHenry Library
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