As on arXiv
Abstract
We consider translators (i.e., initial condition of translating solitons) to mean curvature flow (MCF) in the hyperbolic -space , providing existence and classification results. More specifically, we show the existence and uniqueness of two distinct one-parameter families of complete rotational translators in , one containing catenoid-type translators, and the other parabolic cylindrical ones. We establish a tangency principle for translators in and apply it to prove that properly immersed translators to MCF in are not cylindrically bounded. As a further application of the tangency principle, we prove that any horoconvex translator which is complete or transversal to the -axis is necessarily an open set of a horizontal horosphere. In addition, we classify all translators in which have constant mean curvature. We also consider rotators (i.e., initial condition of rotating solitons) to MCF in and, after classifying the rotators of constant mean curvature, we show that there exists a one-parameter family of complete rotators which are all helicoidal, bringing to the hyperbolic context a distinguished result by Halldorsson, set in .
Comments: Refined proofs from previous version
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Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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Rotators-Translators to Mean Curvature Flow in
Topics Self-shrinkers & translators · Same authors R. F. de Lima, A. K. Ramos, J. P. dos Santos · Shared terms “parameter family”, “helicoidal”
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Translators to Higher Order Mean Curvature Flows in and
Topics Self-shrinkers & translators, Stability & uniqueness · Same author R. F. de Lima · Shared terms “MCF”, “type translators”, “catenoid type”
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Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space
Topics Self-shrinkers & translators, Stability & uniqueness · Same authors R. F. de Lima, A. K. Ramos · Shared terms “MCF”, “translating solitons”, “horosphere”
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Uniqueness of the -catenary cylinders by their asymptotic behaviour
Topics Self-shrinkers & translators, Stability & uniqueness · Same author J. P. dos Santos · Shared terms “translating solitons”, “properly”
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Translators of the Mean Curvature Flow in Hyperbolic Einstein's Static Universe
Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “horosphere”
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Classification of Semigraphical Translators
Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “Specifically”
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Half-space theorems for translating solitons of the r-mean curvature flow
Topics Self-shrinkers & translators · Shared terms “translating solitons”, “properly immersed”, “transversal”
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Poincaré inequality and topological rigidity of translators and self-expanders for the mean curvature flow
Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “properly immersed”
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Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.