STABILITY AND TOPOLOGY OF TRANSLATING SOLITONS FOR THE MEAN CURVATURE FLOW WITH THE SMALL Lm NORM OF THE SECOND FUNDAMENTAL FORM
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In this paper, we show that a complete translating soliton $\Sigma^m$ in $\mathbb R^n$ for the mean curvature flow is stable with respect to weighted volume functional if $\Sigma$ satisfies that the $L^m$ norm of the second fundamental form is smaller than an explicit constant that depends only on the dimension of $\Sigma$ and the Sobolev constant provided in Michael and Simon \cite{Michael & Simon}. Under the same assumption, we also prove that under this upper bound, there is no non-trivial $f$-harmonic $1$-form of $L^2_f$ on $\Sigma$. With the additional assumption that $\Sigma$ is contained in an upper half-space with respect to the translating direction then it has only one end.
Abstract
In this paper, we show that a complete translating soliton $\Sigma^m$ in $\mathbb R^n$ for the mean curvature flow is stable with respect to weighted volume functional if $\Sigma$ satisfies that the $L^m$ norm of the second fundamental form is smaller than an explicit constant that depends only on the dimension of $\Sigma$ and the Sobolev constant provided in Michael and Simon \cite{Michael & Simon}. Under the same assumption, we also prove that under this upper bound, there is no non-trivial $f$-harmonic $1$-form of $L^2_f$ on $\Sigma$. With the additional assumption that $\Sigma$ is contained in an upper half-space with respect to the translating direction then it has only one end.
참고문헌 (20)
- Steven J. Altschuler, "Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle", Calculus of Variations and Partial Differential Equations, 2(1), pp.101~111 (1994), Springer Science and Business Media LLC[DOI]
- Huai-Dong Cao, "The structure of stable minimal hypersurfaces in ℝn+1", Mathematical Research Letters, 4(5), pp.637~644 (1997), International Press of Boston[DOI]
- Julie Clutterbuck, "Stability of translating solutions to mean curvature flow", Calculus of Variations and Partial Differential Equations, 29(3), pp.281~293 (2007), Springer Science and Business Media LLC[DOI]
- D. Hoffman, "Graphical translators for mean curvature flow", Calculus of Variations and Partial Differential Equations, 58(4) (2019), Springer Science and Business Media LLC[DOI]
- Debora Impera, "Rigidity results and topology at infinity of translating solitons of the mean curvature flow", Communications in Contemporary Mathematics, 19(06), pp.1750002~ (2017), World Scientific Pub Co Pte Lt[DOI]
- Debora Impera, "Quantitative index bounds for translators via topology", Mathematische Zeitschrift, 292(1-2), pp.513~527 (2019), Springer Science and Business Media LLC[DOI]
- Daehwan Kim, "Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow", Discrete & Continuous Dynamical Systems - A, 38(11), pp.5897~5919 (2018), American Institute of Mathematical Sciences (AIMS)[DOI]
- Daehwan Kim, "Half-space type theorem for translating solitons of the mean curvature flow in Euclidean space", Proceedings of the American Mathematical Society, Series B, 8(1), pp.1~10 (2021), American Mathematical Society (AMS)[DOI]
- Keita Kunikawa, "Bernstein-type theorem of translating solitons in arbitrary codimension with flat normal bundle", Calculus of Variations and Partial Differential Equations, 54(2), pp.1331~1344 (2015), Springer Science and Business Media LLC[DOI]
- Keita Kunikawa, "Translating solitons in arbitrary codimension", Asian Journal of Mathematics, 21(5), pp.855~872 (2017), International Press of Boston[DOI]
- Keita Kunikawa, "Remarks on topology of stable translating solitons", Geometriae Dedicata, 202(1), pp.1~8 (2018), Springer Science and Business Media LLC[DOI]
- J. H. Michael, "Sobolev and mean-value inequalities on generalized submanifolds of ℝn", Communications on Pure and Applied Mathematics, 26(3), pp.361~379 (1973), Wiley[DOI]
- R. Miyaoka, "Geometry and global analysis", pp.289~293 (1993), Tohoku Univ
- Lei Ni, "Gap theorems for minimal submanifolds in ℝn+1", Communications in Analysis and Geometry, 9(3), pp.641~656 (2001), International Press of Boston[DOI]
- Bennett Palmer, "Stability of minimal hypersurfaces", Commentarii Mathematici Helvetici, 66(1), pp.185~188 (1991), European Mathematical Society - EMS - Publishing House GmbH[DOI]
- Keomkyo Seo, "Minimal submanifolds with small total scalar curvature in Euclidean space", Kodai Mathematical Journal, 31(1) (2008), Tokyo Institute of Technology, Department of Mathematics[DOI]
- Yi-Bing Shen, "On stable complete minimal hypersurfaces in R n +1", American Journal of Mathematics, 120(1), pp.103~116 (1998), Project MUSE[DOI]
- Joel Spruck, "Remarks on the stability of minimal submanifolds of IR n", Mathematische Zeitschrift, 144(2), pp.169~174 (1975), Springer Science and Business Media LLC[DOI]
- Qiaoling Wang, "On minimal submanifolds in an Euclidean space", Mathematische Nachrichten, 261-262(1), pp.176~180 (2003), Wiley[DOI]
- Y. L. Xin, "Translating solitons of the mean curvature flow", Calculus of Variations and Partial Differential Equations, 54(2), pp.1995~2016 (2015), Springer Science and Business Media LLC[DOI]