161 research outputs found
Non-existence for self-translating solitons
This paper establishes geometric obstructions to the existence of complete,
properly embedded, mean curvature flow self-translating solitons
, generalizing previously known
non-existence conditions such as cylindrical boundedness.Comment: 15 page
Mean curvature self-shrinkers of high genus: Non-compact examples
We give the first rigorous construction of complete, embedded self-shrinking
hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The
surfaces exist for any sufficiently large prescribed genus , and are
non-compact with one end. Each has symmetries and comes from
desingularizing the intersection of the plane and sphere through a great
circle, a configuration with very high symmetry. Each is at infinity asymptotic
to the cone in over a -periodic graph on an equator
of the unit sphere , with the shape of a
periodically "wobbling sheet". This is a dramatic instability phenomenon, with
changes of asymptotics that break much more symmetry than seen in minimal
surface constructions. The core of the proof is a detailed understanding of the
linearized problem in a setting with severely unbounded geometry, leading to
special PDEs of Ornstein-Uhlenbeck type with fast growth on coefficients of the
gradient terms. This involves identifying new, adequate weighted H\"older
spaces of asymptotically conical functions in which the operators invert, via a
Liouville-type result with precise asymptotics.Comment: 41 pages, 1 figure; minor typos fixed; to appear in J. Reine Angew.
Mat
Ancient mean curvature flows and their spacetime tracks
We study properly immersed ancient solutions of the codimension one mean
curvature flow in -dimensional Euclidean space, and classify the convex
hulls of the subsets of space reached by any such flow. In particular, it
follows that any compact convex ancient mean curvature flow can only have a
slab, a halfspace or all of space as the closure of its set of reach. The proof
proceeds via a bi-halfspace theorem (also known as a wedge theorem) for ancient
solutions derived from a parabolic Omori-Yau maximum principle for ancient mean
curvature flows.Comment: 14 pages; minor edits, added proof details and reference
Mean curvature flow self-shrinkers with genus and asymptotically conical ends
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 121-124).This doctoral dissertation is on the theory of Minimal Surfaces and of singularities in Mean Curvature Flow, for smooth submanifolds Y" in an ambient Riemannian (n+ 1)-manifold Nn+1, including: (1) New asymptotically conical self-shrinkers with a symmetry, in R"+1. (1') Classification of complete embedded self-shrinkers with a symmetry, in IR"+1, and of asymptotically conical ends with a symmetry. (2) Construction of complete, embedded self-shrinkers E2 C R3 of genus g, with asymptotically conical infinite ends, via minimal surface gluing. (3) Construction of closed embedded self-shrinkers y2 C R3 with genus g, via minimal surface gluing. In the work there are two central geometric and analytic themes that cut across (1)-(3): The notion of asymptotically conical infinite ends in (1)-(1') and (2), and in (2) and (3) the gluing methods for minimal surfaces which were developed by Nikolaos Kapouleas. For the completion of (2) it was necessary to initiate the development of a stability theory in a setting with unbounded geometry, the manifolds in question having essentially singular (worse than cusp-like) infinities. This was via a Schauder theory in weighted Hölder spaces for the stability operator, which is a Schrodinger operator of Ornstein-Uhlenbeck type, on the self-shrinkers viewed as minimal surfaces. This material is, for the special case of graphs over the plane, included as part of the thesis. The results in (1)-(1') are published as the joint work [KMø 1] with Stephen Kleene, and the result in (2) was proven in collaboration with Kleene-Kapouleas, and appeared in [KKMø 0]. The results in (3) are contained in the preprint [Mø1].by Niels Martin Moøller.Ph.D
Rigidity and non-existence results for collapsed translators
We prove a rigidity result for mean curvature self-translating solitons,
characterizing the grim reaper cylinder as the only finite entropy
self-translating 2-surface in of width and bounded from
below. The proof makes use of parabolicity in a weighted setting applied to a
suitable universally -superharmonic function defined on translaters in such
slabs.Comment: Expanded introduction. New result added (Theorem 1.6). All comments
are welcome
Entropy Bounds, Compactness and Finiteness Theorems for Embedded Self-shrinkers with Rotational Symmetry
In this work, we study the space of complete embedded rotationally symmetric
self-shrinking hypersurfaces in . First, using comparison
geometry in the context of metric geometry, we derive explicit upper bounds for
the entropy of all such self-shrinkers. Second, as an application we prove a
smooth compactness theorem on the space of all such shrinkers. We also prove
that there are only finitely many such self-shrinkers with an extra reflection
symmetry.Comment: Accepted for publication in Crelle. Accepted version, updated to
reflect referee remark
The distribution of quantum fidelities
When applied to different input states, an imperfect quantum operation yields
output states with varying fidelities, defined as the absolute square of their
overlap with the desired states. We present an expression for the distribution
of fidelities for a class of operations applied to a general qubit state, and
we present general expressions for the variance and input-space averaged
fidelities of arbitrary linear maps on finite dimensional Hilbert spaces.Comment: 5 pages, 1 figur
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