24 research outputs found
Asymptotically hyperbolic normalized Ricci flow and rotational symmetry
We consider the normalized Ricci flow evolving from an initial metric which
is conformally compactifiable and asymptotically hyperbolic. We show that there
is a unique evolving metric which remains in this class, and that the flow
exists up to the time where the norm of the Riemann tensor diverges.
Restricting to initial metrics which belong to this class and are rotationally
symmetric, we prove that if the sectional curvature in planes tangent to the
orbits of symmetry is initially nonpositive, the flow starting from such an
initial metric exists for all time. Moreover, if the sectional curvature in
planes tangent to these orbits is initially negative, the flow converges at an
exponential rate to standard hyperbolic space. This restriction on sectional
curvature automatically rules out initial data admitting a minimal hypersphere.Comment: 28 pages, replaced one-word error in two locations on page
Yamabe flow on manifolds with edges
Let (M,g) be a compact oriented Riemannian manifold with an incomplete edge
singularity. This article shows that it is possible to evolve g by the Yamabe
flow within a class of singular edge metrics. As the main analytic step we
establish parabolic Schauder-type estimates for the heat operator on certain
H\"older spaces adapted to the singular edge geometry. We apply these estimates
to obtain local existence for a variety of quasilinear equations, including the
Yamabe flow. This provides a setup for a subsequent discussion of the Yamabe
problem using flow techniques in the singular setting.Comment: 3 pages, 1 figure, v2: minor improvements, references added,
organizational change
H\"older Compactification for some manifolds with pinched negative curvature near infinity
We consider a complete noncompact Riemannian manifold M and give conditions
on a compact submanifold K of M so that the outward normal exponential map off
of the boundary of K is a diffeomorphism onto M\K. We use this to compactify M
and show that pinched negative sectional curvature outside K implies M has a
compactification with a well defined H\"older structure independent of K. The
H\"older constant depends on the ratio of the curvature pinching. This extends
and generalizes a 1985 result of Anderson and Schoen.Comment: 27 pages, 1 figur
Renormalized volume and the evolution of APEs
We study the evolution of the renormalized volume functional for
asymptotically Poincare-Einstein metrics (M,g) which are evolving by normalized
Ricci flow. In particular, we prove that the time derivative of the
renormalized volume along the flow is the negative integral of scal(g(t)) +
n(n-1) over the manifold. This implies that if scal(g(0))+n(n-1) is
non-negative at t=0, then the renormalized volume decreases monotonically.
We also discuss how, when n=4, our results describe the Hawking-Page phase
transition. Differences in renormalized volumes give rigorous meaning to the
Hawking-Page difference of actions and describe the free energy liberated in
the transition.Comment: 15 pages; errors corrected and section 4 modifie
Short-time existence for some higher-order geometric flows
We establish short-time existence and regularity for higher-order flows
generated by a class of polynomial natural tensors that, after an adjustment by
the Lie derivative of the metric with respect to a suitable vector field, have
strongly parabolic linearizations. We apply this theorem to flows by powers of
the Laplacian of the Ricci tensor, and to flows generated by the ambient
obstruction tensor. As a special case, we prove short-time existence for a type
of Bach flow.Comment: 17 page
Conformal compactification of asymptotically locally hyperbolic metrics
In this paper we study the extent to which conformally compact asymptotically
hyperbolic metrics may be characterized intrinsically. Building on the work of
the first author, we prove that decay of sectional curvature to -1 and decay of
covariant derivatives of curvature outside an appropriate compact set yield
H\"older regularity for a conformal compactification of the metric. In the
Einstein case, we prove that the estimate on the sectional curvature implies
the control of all covariant derivatives of the Weyl tensor, permitting us to
strengthen our result
Long-time existence of the edge Yamabe flow
This article presents an analysis of the normalized Yamabe flow starting at
and preserving a class of compact Riemannian manifolds with incomplete edge
singularities and negative Yamabe invariant. Our main results include
uniqueness, long-time existence and convergence of the edge Yamabe flow
starting at a metric with everywhere negative scalar curvature. Our methods
include novel maximum principle results on the singular edge space without
using barrier functions. Moreover, our uniform bounds on solutions are
established by a new ansatz without in any way using or redeveloping
Krylov-Safonov estimates in the singular setting. As an application we obtain a
solution to the Yamabe problem for incomplete edge metrics with negative Yamabe
invariant using flow techniques. Our methods lay groundwork for studying other
flows like the mean curvature flow as well as the porous medium equation in the
singular setting.Comment: 38 page
Uniqueness for Some Higher-Order Geometric Flows
We show that solutions to certain higher-order intrinsic geometric flows on a
compact manifold, including some flows generated by the ambient obstruction
tensor, are unique. With the goal of providing a complete self-contained proof,
details surrounding map covariant derivatives and a careful application of the
DeTurck trick are provided.Comment: 15 page
Ricci Flow and Volume Renormalizability
With respect to any special boundary defining function, a conformally compact
asymptotically hyperbolic metric has an asymptotic expansion near its conformal
infinity. If this expansion is even to a certain order and satisfies one extra
condition, then it is possible to define its renormalized volume and show that
it is independent of choices that preserve this evenness structure. We prove
that such expansions are preserved under normalized Ricci flow. We also study
the variation of curvature functionals in this setting, and as one application,
obtain the variation formula where is the scalar curvature for the evolving metric
, and is Riesz
renormalization. This extends our earlier work to a broader class of metrics.Comment: 21 page
Intrinsic characterization for Lipschitz asymptotically hyperbolic metrics
Conformally compact asymptotically hyperbolic metrics have been intensively
studied. The goal of this note is to understand what intrinsic conditions on a
complete Riemannian manifold (M,g) will ensure that g is asymptotically
hyperbolic in this sense. We use the geodesic compactification by asymptotic
geodesic rays to compactify M and appropriate curvature decay conditions to
study the regularity of the conformal compactification. We also present an
interesting example that shows our conclusion is nearly optimal for our
assumptions.Comment: 18 pages; correction to Theorem