Details
| Originalsprache | Englisch |
|---|---|
| Seiten (von - bis) | 433-445 |
| Seitenumfang | 13 |
| Fachzeitschrift | Vietnam Journal of Mathematics |
| Jahrgang | 49 |
| Ausgabenummer | 2 |
| Frühes Online-Datum | 12 Jan. 2021 |
| Publikationsstatus | Veröffentlicht - Juni 2021 |
Abstract
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Allgemeine Mathematik
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in: Vietnam Journal of Mathematics, Jahrgang 49, Nr. 2, 06.2021, S. 433-445.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Self-Expanders of the Mean Curvature Flow
AU - Smoczyk, Knut
N1 - Funding Information: The author was supported by the German Research Foundation within the priority program SPP 2026 - Geometry at Infinity, DFG SM 78/7-1.
PY - 2021/6
Y1 - 2021/6
N2 - We study self-expanding solutions Mm ⊂ Rn of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of selfexpanding curves and flat subspaces, if and only if the function |A|2/|H|2 attains a localmaximum, where A denotes the second fundamental form and H the mean curvature vectorof M. If the principal normal ξ = H/|H| is parallel in the normal bundle, then a similar result holds in higher codimension for the function |Aξ |2/|H|2, where Aξ is the second fundamental form with respect to ξ . As a corollary we obtain that complete mean convex self-expanders attain strictly positive scalar curvature, if they are smoothly asymptotic to cones of non-negative scalar curvature. In particular, in dimension 2 any mean convex selfexpander that is asymptotic to a cone must be strictly convex.
AB - We study self-expanding solutions Mm ⊂ Rn of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of selfexpanding curves and flat subspaces, if and only if the function |A|2/|H|2 attains a localmaximum, where A denotes the second fundamental form and H the mean curvature vectorof M. If the principal normal ξ = H/|H| is parallel in the normal bundle, then a similar result holds in higher codimension for the function |Aξ |2/|H|2, where Aξ is the second fundamental form with respect to ξ . As a corollary we obtain that complete mean convex self-expanders attain strictly positive scalar curvature, if they are smoothly asymptotic to cones of non-negative scalar curvature. In particular, in dimension 2 any mean convex selfexpander that is asymptotic to a cone must be strictly convex.
KW - Mean curvature flow
KW - Self-expander
UR - http://www.scopus.com/inward/record.url?scp=85110809248&partnerID=8YFLogxK
U2 - 10.1007/s10013-020-00469-1
DO - 10.1007/s10013-020-00469-1
M3 - Article
VL - 49
SP - 433
EP - 445
JO - Vietnam Journal of Mathematics
JF - Vietnam Journal of Mathematics
IS - 2
ER -