42 research outputs found
Ancient solutions of geometric flows with curvature pinching
We prove rigidity theorems for ancient solutions of geometric flows of
immersed submanifolds. Specifically, we find pinching conditions on the second
fundamental form that characterize the shrinking sphere among compact ancient
solutions for the mean curvature flow in codimension greater than one, and for
some nonlinear curvature flows of hypersurfaces.Comment: Minor changes, bibliography updated. To appear on The Journal of
Geometric Analysis. The final version is available online at
https://doi.org/10.1007/s12220-018-0036-
Mean curvature flow of pinched submanifolds of
We consider the evolution by mean curvature flow of a closed submanifold of
the complex projective space. We show that, if the submanifold has small
codimension and satisfies a suitable pinching condition on the second
fundamental form, then the evolution has two possible behaviors: either the
submanifold shrinks to a round point in finite time, or it converges smoothly
to a totally geodesic limit in infinite time. The latter behavior is only
possible if the dimension is even. These results generalize previous works by
Huisken and Baker on the mean curvature flow of submanifolds of the sphere.Comment: 39 pages, minor correction
Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes
We consider the mean curvature flow of a closed hypersurface in the complex
or quaternionic projective space. Under a suitable pinching assumption on the
initial data, we prove apriori estimates on the principal curvatures which
imply that the asymptotic profile near a singularity is either strictly convex
or cylindrical. This result generalizes to a large class of symmetric ambient
spaces the estimates obtained in the previous works on the mean curvature flow
of hypersurfaces in Euclidean space and in the sphere.Comment: 17 pages, minor correction
Volume preserving flow by powers of symmetric polynomials in the principal curvatures
We study a volume preserving curvature flow of convex hypersurfaces, driven
by a power of the -th elementary symmetric polynomial in the principal
curvatures. Unlike most of the previous works on related problems, we do not
require assumptions on the curvature pinching of the initial datum. We prove
that the solution exists for all times and that the speed remains bounded and
converges to a constant in an integral norm. In the case of the volume
preserving scalar curvature flow, we can prove that the hypersurfaces converge
smoothly and exponentially fast to a round sphere.Comment: 17 page
Global Propagation of Singularities for Time Dependent Hamilton-Jacobi Equations
We investigate the properties of the set of singularities of semiconcave
solutions of Hamilton-Jacobi equations of the form \begin{equation*}
u_t(t,x)+H(\nabla u(t,x))=0, \qquad\text{a.e. }(t,x)\in
(0,+\infty)\times\Omega\subset\mathbb{R}^{n+1}\,. \end{equation*} It is well
known that the singularities of such solutions propagate locally along
generalized characteristics. Special generalized characteristics, satisfying an
energy condition, can be constructed, under some assumptions on the structure
of the Hamiltonian . In this paper, we provide estimates of the dissipative
behavior of the energy along such curves. As an application, we prove that the
singularities of any viscosity solution of the above equation cannot vanish in
a finite time
Neckpinch singularities in fractional mean curvature flows
In this paper we consider the evolution of sets by a fractional mean
curvature flow. Our main result states that for any dimension , there
exists an embedded surface in evolving by fractional mean
curvature flow, which developes a singularity before it can shrink to a point.
When this result generalizes the analogue result of Grayson for the
classical mean curvature flow. Interestingly, when , our result provides
instead a counterexample in the nonlocal framework to the well known Grayson
Theorem, which states that any smooth embedded curve in the plane evolving by
(classical) MCF shrinks to a point
Neckpinch singularities in fractional mean curvature flows
In this paper we consider the evolution of boundaries of sets by a fractional mean curvature flow.
We show that, for any dimension n ≥ 2, there exist embedded hypersurfaces in \Rn which develop a
singularity without shrinking to a point. Such examples are well known for the classical mean curvature flow
for n ≥ 3. Interestingly, when n=2, our result provides instead a counterexample in the nonlocal
framework to the well known Grayson's Theorem [17], which states that any smooth embedded
curve in the plane evolving by (classical) MCF shrinks to a point. The essential step in our construction is an
estimate which ensures that a suitably small perturbation of a thin strip has positive fractional curvature at
every boundary point