Parabolicity, Brownian Exit Time and Properness of Solitons of the Direct and Inverse Mean Curvature Flow
We study some potential theoretic properties of homothetic solitons Σ n of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in R n + m , we observe similarities and differences in the geometry of solitons in both flows. In particular, we show...
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| Veröffentlicht in: | The Journal of Geometric Analysis 2021, Vol.31 (1), p.579-618 |
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| container_title | The Journal of Geometric Analysis |
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| creator | Gimeno, Vicent Palmer, Vicente |
| description | We study some potential theoretic properties of homothetic solitons
Σ
n
of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in
R
n
+
m
, we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that parabolic MCF-solitons
Σ
n
with
n
>
2
are self-shrinkers and that parabolic IMCF-solitons of any dimension are self-expanders. We have studied too the geometric behavior of parabolic MCF and IMCF-solitons confined in a ball, the behavior of the mean exit time function for the Brownian motion defined on
Σ
as well as a classification of properly immersed MCF-self-shrinkers with bounded second fundamental form, following the lines of Cao and Li (Calc Var 46:879–889, 2013). |
| doi_str_mv | 10.1007/s12220-019-00291-3 |
| format | Article |
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Σ
n
of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in
R
n
+
m
, we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that parabolic MCF-solitons
Σ
n
with
n
>
2
are self-shrinkers and that parabolic IMCF-solitons of any dimension are self-expanders. We have studied too the geometric behavior of parabolic MCF and IMCF-solitons confined in a ball, the behavior of the mean exit time function for the Brownian motion defined on
Σ
as well as a classification of properly immersed MCF-self-shrinkers with bounded second fundamental form, following the lines of Cao and Li (Calc Var 46:879–889, 2013).</description><identifier>ISSN: 1050-6926</identifier><identifier>EISSN: 1559-002X</identifier><identifier>DOI: 10.1007/s12220-019-00291-3</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Abstract Harmonic Analysis ; Brownian motion ; Convex and Discrete Geometry ; Differential Geometry ; Dynamical Systems and Ergodic Theory ; Expanders ; Fourier Analysis ; Geometry ; Global Analysis and Analysis on Manifolds ; Manifolds (mathematics) ; Mathematics ; Mathematics and Statistics ; Solitary waves ; Time functions</subject><ispartof>The Journal of Geometric Analysis, 2021, Vol.31 (1), p.579-618</ispartof><rights>Mathematica Josephina, Inc. 2019</rights><rights>COPYRIGHT 2021 Springer</rights><rights>Mathematica Josephina, Inc. 2019.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c358t-f8cc3c433997422d8375283941e0ca14692770b6010a6487ee1cef824c290b3</citedby><cites>FETCH-LOGICAL-c358t-f8cc3c433997422d8375283941e0ca14692770b6010a6487ee1cef824c290b3</cites><orcidid>0000-0002-0033-7717 ; 0000-0001-7982-1677</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s12220-019-00291-3$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s12220-019-00291-3$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Gimeno, Vicent</creatorcontrib><creatorcontrib>Palmer, Vicente</creatorcontrib><title>Parabolicity, Brownian Exit Time and Properness of Solitons of the Direct and Inverse Mean Curvature Flow</title><title>The Journal of Geometric Analysis</title><addtitle>J Geom Anal</addtitle><description>We study some potential theoretic properties of homothetic solitons
Σ
n
of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in
R
n
+
m
, we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that parabolic MCF-solitons
Σ
n
with
n
>
2
are self-shrinkers and that parabolic IMCF-solitons of any dimension are self-expanders. We have studied too the geometric behavior of parabolic MCF and IMCF-solitons confined in a ball, the behavior of the mean exit time function for the Brownian motion defined on
Σ
as well as a classification of properly immersed MCF-self-shrinkers with bounded second fundamental form, following the lines of Cao and Li (Calc Var 46:879–889, 2013).</description><subject>Abstract Harmonic Analysis</subject><subject>Brownian motion</subject><subject>Convex and Discrete Geometry</subject><subject>Differential Geometry</subject><subject>Dynamical Systems and Ergodic Theory</subject><subject>Expanders</subject><subject>Fourier Analysis</subject><subject>Geometry</subject><subject>Global Analysis and Analysis on Manifolds</subject><subject>Manifolds (mathematics)</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Solitary waves</subject><subject>Time functions</subject><issn>1050-6926</issn><issn>1559-002X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kUtLAzEUhQdRsD7-gKuAW0dvHjOZWWp9gmKhLtyFNL1TI21Sk7Taf2_sCO4ki9xcvnNPLqcoTiicUwB5ESljDEqgbQnAWlrynWJAq2r7fN3NNVRQ1i2r94uDGN8BRM2FHBR2pIOe-Lk1Nm3OyFXwn85qR26-bCIvdoFEuykZBb_E4DBG4jsyznjyblunNyTXNqBJW_DBrTFEJE-YZwxXYa3TKiC5nfvPo2Kv0_OIx7_3YTG-vXkZ3pePz3cPw8vH0vCqSWXXGMON4LxtpWBs2nBZsYa3giIYTUXeQUqY1EBB16KRiNRg1zBhWAsTflic9lOXwX-sMCb17lfBZUPFhGwbEKzmmTrvqZmeo7Ku8ylok88UF9Z4h53N_UsJkotsJLKA9QITfIwBO7UMdqHDRlFQPwmoPgGVE1DbBNSPC-9FMcNuhuHvL_-ovgEUiYdd</recordid><startdate>2021</startdate><enddate>2021</enddate><creator>Gimeno, Vicent</creator><creator>Palmer, Vicente</creator><general>Springer US</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>IAO</scope><orcidid>https://orcid.org/0000-0002-0033-7717</orcidid><orcidid>https://orcid.org/0000-0001-7982-1677</orcidid></search><sort><creationdate>2021</creationdate><title>Parabolicity, Brownian Exit Time and Properness of Solitons of the Direct and Inverse Mean Curvature Flow</title><author>Gimeno, Vicent ; Palmer, Vicente</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c358t-f8cc3c433997422d8375283941e0ca14692770b6010a6487ee1cef824c290b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Abstract Harmonic Analysis</topic><topic>Brownian motion</topic><topic>Convex and Discrete Geometry</topic><topic>Differential Geometry</topic><topic>Dynamical Systems and Ergodic Theory</topic><topic>Expanders</topic><topic>Fourier Analysis</topic><topic>Geometry</topic><topic>Global Analysis and Analysis on Manifolds</topic><topic>Manifolds (mathematics)</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Solitary waves</topic><topic>Time functions</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Gimeno, Vicent</creatorcontrib><creatorcontrib>Palmer, Vicente</creatorcontrib><collection>CrossRef</collection><collection>Gale Academic OneFile</collection><jtitle>The Journal of Geometric Analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Gimeno, Vicent</au><au>Palmer, Vicente</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Parabolicity, Brownian Exit Time and Properness of Solitons of the Direct and Inverse Mean Curvature Flow</atitle><jtitle>The Journal of Geometric Analysis</jtitle><stitle>J Geom Anal</stitle><date>2021</date><risdate>2021</risdate><volume>31</volume><issue>1</issue><spage>579</spage><epage>618</epage><pages>579-618</pages><issn>1050-6926</issn><eissn>1559-002X</eissn><abstract>We study some potential theoretic properties of homothetic solitons
Σ
n
of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in
R
n
+
m
, we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that parabolic MCF-solitons
Σ
n
with
n
>
2
are self-shrinkers and that parabolic IMCF-solitons of any dimension are self-expanders. We have studied too the geometric behavior of parabolic MCF and IMCF-solitons confined in a ball, the behavior of the mean exit time function for the Brownian motion defined on
Σ
as well as a classification of properly immersed MCF-self-shrinkers with bounded second fundamental form, following the lines of Cao and Li (Calc Var 46:879–889, 2013).</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s12220-019-00291-3</doi><tpages>40</tpages><orcidid>https://orcid.org/0000-0002-0033-7717</orcidid><orcidid>https://orcid.org/0000-0001-7982-1677</orcidid></addata></record> |
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| ispartof | The Journal of Geometric Analysis, 2021, Vol.31 (1), p.579-618 |
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| language | eng |
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| subjects | Abstract Harmonic Analysis Brownian motion Convex and Discrete Geometry Differential Geometry Dynamical Systems and Ergodic Theory Expanders Fourier Analysis Geometry Global Analysis and Analysis on Manifolds Manifolds (mathematics) Mathematics Mathematics and Statistics Solitary waves Time functions |
| title | Parabolicity, Brownian Exit Time and Properness of Solitons of the Direct and Inverse Mean Curvature Flow |
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