# Sharp rigidity estimates for nearly umbilical surfaces

created on 22 May 2003
modified by delellis on 03 May 2011

[BibTeX]

Published Paper

Inserted: 22 may 2003
Last Updated: 3 may 2011

Journal: Journal of Differential Geometry
Volume: 69
Number: 75--110
Year: 2003

Abstract:

A classical theorem in differential geometry states that if $\Sigma\subset *R*^3$ is a compact connected surface without boundary and all points of $\Sigma$ are umbilical, then $\Sigma$ is a round sphere and therefore its second fundamental form $A$ is a constant multiple of the identity. In this paper we give a sharp quantitative version of this theorem. More precisely we prove that if the $L^2$ norm of the traceless part of $A$ is small, then $A$ is $L^2$ near to a constant multiple of the identity.

For the most updated version and eventual errata see the page

http:/www.math.uzh.chindex.php?id=publikationen&key1=493

Credits | Cookie policy | HTML 5 | CSS 2.1
{"ft_lang_label":"__label__en","ft_lang_prob":0.7937791,"math_prob":0.9665141,"size":842,"snap":"2021-31-2021-39","text_gpt3_token_len":217,"char_repetition_ratio":0.090692125,"word_repetition_ratio":0.032258064,"special_character_ratio":0.24703088,"punctuation_ratio":0.12883435,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98017603,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-27T09:15:36Z\",\"WARC-Record-ID\":\"<urn:uuid:d2f288db-0da9-4f41-98cf-8675b8ca1401>\",\"Content-Length\":\"4619\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d6006393-9ff1-47a4-85e8-239d2d8fd975>\",\"WARC-Concurrent-To\":\"<urn:uuid:22461185-ff0a-45dc-be84-c16bb11e0b21>\",\"WARC-IP-Address\":\"192.167.206.42\",\"WARC-Target-URI\":\"https://cvgmt.sns.it/paper/1306/\",\"WARC-Payload-Digest\":\"sha1:G44E7F5N3TDIDU66ZXJ5UD5TKRAM72MI\",\"WARC-Block-Digest\":\"sha1:6L3NAT6AT3PXW25HI6XO76UUF4ILMQ7I\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780058415.93_warc_CC-MAIN-20210927090448-20210927120448-00665.warc.gz\"}"}