Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Two-sided non-collapsing curvature flows   It was recently proved that embedded solutions of Euclidean hypersurface
flows with speeds given by concave (convex), degree one homogeneous functions
of the Weingarten map are interior (exterior) non-collapsing. These results
were subsequently extended to hypersurface flows in the sphere and hyperbolic
space.
  In the first part of the paper, we show that locally convex solutions are
exterior non-collapsing for a larger class of speed functions than previously
considered; more precisely, we show that the previous results hold when
convexity of the speed function is relaxed to inverse-concavity. We note that
inverse-concavity is satisfied by a large class of concave speed functions. As
a consequence, we obtain a large class of two-sided non-collapsing flows,
whereas previously two-sided non-collapsing was only known for the mean
curvature flow.
  In the second part of the paper, we demonstrate the utility of two sided
non-collapsing with a straightforward proof of convergence of compact, convex
hypersurfaces to round points.
  The proof of the non-collapsing estimate is similar to the previous results
mentioned, in that we show that the exterior ball curvature is a viscosity
supersolution of the linearised flow. The new ingredient is the following
observation: Since the function which provides an upper support in the
derivation of the viscosity inequality is defined on $M\times M$ (or $TM$ in
the `boundary case'), whereas the exterior ball curvature and the linearised
flow equation depend only on the first factor, we are privileged with a freedom
of choice in which second derivatives from the extra directions to include in
the calculation. The optimal choice is closely related to the class of
inverse-concave speed functions.

5
The categories of this paper are: gr-qc
15
The categories of this paper are: nucl-ex
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The categories of this paper are: cs
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The categories of this paper are: physics
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The categories of this paper are: nucl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: econ
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The categories of this paper are: math-ph
31
The categories of this paper are: ao-sci
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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: eess
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-th
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The categories of this paper are: acc-phys
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The categories of this paper are: dg-ga