Details
| Original language | English |
|---|---|
| Pages (from-to) | 433-445 |
| Number of pages | 13 |
| Journal | Vietnam Journal of Mathematics |
| Volume | 49 |
| Issue number | 2 |
| Early online date | 12 Jan 2021 |
| Publication status | Published - Jun 2021 |
Abstract
Cite this
- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS
In: Vietnam Journal of Mathematics, Vol. 49, No. 2, 06.2021, p. 433-445.
Research output: Contribution to journal › Article › Research › 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 -