Entropy bounds, compactness and finiteness theorems for embedded self-shrinkers with rotational symmetry

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In this work, we study the space of complete embedded rotationally symmetric self-shrinking hypersurfaces in ℝ n + 1 {\mathbb{R}^{n+1}}. 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.

Original languageEnglish
JournalJournal fur die Reine und Angewandte Mathematik
Pages (from-to)239-259
Publication statusPublished - 2022

Bibliographical note

Publisher Copyright:
© 2022 Walter de Gruyter GmbH, Berlin/Boston 2022 Independent Research Fund Denmark DFF Sapere Aude 7027-00110B Danish National Research Foundation CPH-GEOTOP-DNRF151 Carlsberg Foundation CF21-0680 The authors were partially supported by DFF Sapere Aude 7027-00110B, by CPH-GEOTOP-DNRF151 and by CF21-0680 from respectively the Independent Research Fund Denmark, the Danish National Research Foundation and the Carlsberg Foundation.

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