24,690 research outputs found
Minimizer of an isoperimetric ratio on a metric on with finite total area
Let be a complete Riemmanian metric on with finite total
area and with
where
is any closed simple curve in , is the length of ,
and are the area of the regions inside and
outside respectively, with respect to the metric . We prove the
existence of a minimizer for . As a corollary we obtain a new proof for
the existence of a minimizer for for any when the metric
is the maximal solution of the Ricci flow
equation \1 g_{ij}/\1 t=-2R_{ij} on \cite{DH} where
is the extinction time of the solution.Comment: 14 pages, some typos are corrected and some proofs are written in
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Another proof of Ricci flow on incomplete surfaces with bounded above Gauss curvature
We give a simple proof of an extension of the existence results of Ricci flow
of G.Giesen and P.M.Topping [GiT1],[GiT2], on incomplete surfaces with bounded
above Gauss curvature without using the difficult Shi's existence theorem of
Ricci flow on complete non-compact surfaces and the pseudolocality theorem of
G.Perelman [P1] on Ricci flow. We will also give a simple proof of a special
case of the existence theorem of P.M.Topping [T] without using the existence
theorem of W.X.Shi [S1].Comment: 13 page
Uniqueness of solutions of Ricci flow on complete noncompact manifolds
We prove the uniqueness of solutions of the Ricci flow on complete noncompact
manifolds with bounded curvatures using the De Turck approach. As a consequence
we obtain a correct proof of the existence of solution of the Ricci harmonic
flow on complete noncompact manifolds with bounded curvatures.Comment: A simple example of a manifold with bounded curvature and injectivity
radius going to zero as the point tends to infinity is given. A proof and
argument why the crucial lemma Lemma 2.2 of the Chen-Zhu's paper \cite{CZ}
cannot hold is give
A harmonic map flow associated with the standard solution of Ricci flow
Let , , , be a standard solution of
the Ricci flow with radially symmetric initial data . We will extend a
recent existence result of P. Lu and G. Tian and prove that for any there exists a solution of the corresponding harmonic map flow
satisfying of the form in polar coordinates in ,
, where is the radial co-ordinate
with respect to and with
. We will also prove the uniqueness
of solution of the harmonic map flow. We will also use the same technique to
prove that the solution of the heat equation in
has removable singularities at
, , , if and only if
locally uniformly on every compact subset of .Comment: 21 page
A pseudolocality theorem for Ricci flow
In this paper we will give a simple proof of a modification of a result on
pseudolocality for the Ricci flow by P.Lu without using the pseudolocality
theorem 10.1 of Perelman [P1]. We also obtain an extension of a result of
Hamilton on the compactness of a sequence of complete pointed Riemannian
manifolds evolving under Ricci flow with
uniform bounded sectional curvatures on and uniform positive lower
bound on the injectivity radii at with respect to the metric .Comment: 11 pages, I have add one mild assumption on the theorem and
completely rewrites the proof of the theorem which avoids the use of the
logarithmic Sobolev inequality completely. I also obtain an extension of the
compactness result of Hamilton on a sequence of complete pointed Riemannian
manifolds evolving under Ricci flo
Super fast vanishing solutions of the fast diffusion equation
We will extend a recent result of B.Choi, P.Daskalopoulos and J.King. For any
, , we will construct subsolutions
and supersolutions of the fast diffusion equation
in , , which decay at the rate
as . As a consequence we obtain the
existence of unique solution of the Cauchy problem in , in ,
which decay at the rate as when
satisfies appropriate decay condition.Comment: 37 pages, typos corrected, reference update
Generalized \Cal{L}-geodesic and monotonicity of the generalized reduced volume in the Ricci flow
Suppose is a complete n-dimensional manifold, , with a metric
that evolves by the Ricci flow in . For any , , , we define the \Cal{L}_p-length between and
, \Cal{L}_p-geodesic, the generalized reduced distance and the
generalized reduced volume , , corresponding
to the \Cal{L}_p-geodesic at the point at time . Under the
condition on for some
constant , we will prove the existence of a \Cal{L}_p-geodesic which
minimize the \Cal{L}_p(q,\bar{\tau})-length between and for any
. This result for the case is conjectured and used many
times but no proof of it was given in Perelman's papers on Ricci flow. My
result is new and answers in affirmative the existence of such
\Cal{L}-geodesic minimizer for the -length which is crucial to
the proof of many results in Perelman's papers on Ricci flow. We also obtain
many other properties of the generalized \Cal{L}_p-geodesic and generalized
reduced volume.Comment: 64 page
A simple proof on the non-existence of shrinking breathers for the Ricci flow
Suppose is a compact n-dimensional manifold, , with a metric
that evolves by the Ricci flow in
. We will give a simple proof of a recent result of Perelman on
the non-existence of shrinking breather without using the logarithmic Sobolev
inequality.Comment: 15 page
Maximum principle and convergence of fundamental solutions for the Ricci flow
In this paper we will prove a maximum principle for the solutions of linear
parabolic equation on complete non-compact manifolds with a time varying
metric. We will prove the convergence of the Neumann Green function of the
conjugate heat equation for the Ricci flow in to the minimal
fundamental solution of the conjugate heat equation as . We will
prove the uniqueness of the fundamental solution under some exponential decay
assumption on the fundamental solution. We will also give a detail proof of the
convergence of the fundamental solutions of the conjugate heat equation for a
sequence of pointed Ricci flow to the
fundamental solution of the limit manifold as which was used
without proof by Perelman in his proof of the pseudolocality theorem for Ricci
flow.Comment: 15 page
Existence and properties of ancient solutions of the Yamabe flow
Let and . We construct -parameters,
-parameters, -parameters ancient solutions of the equation
, , in for some
. This equation arises in the study of Yamabe flow. We obtain
various properties of the ancient solutions of this equation including exact
decay rate of ancient solutions as . We also prove that both the
-parameters ancient solution and the -parameters ancient solution are
singular limit solution of the -parameters ancient solutions.Comment: 43 pages, some typo corrected, and uniqueness of the 4-paramters
ancient solution adde
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