Evolution of convex hypersurfaces by a fully nonlinear flow

详细信息    查看全文

• 作者：Yan-nan Liu^a ; ^{liuyn@th.btbu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Hong-jie Ju^b
• 关键词：35K45 ; 53A05
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：130
• 期：Complete
• 页码：47-58
• 全文大小：655 K

文摘

In this paper, we study the evolution of convex hypersurfaces by a fully nonlinear function of curvature minus an external force field class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003211&_mathId=si1.gif&_user=111111111&_pii=S0362546X15003211&_rdoc=1&_issn=0362546X&md5=4ff60d1bb3c575bb13422cd75ef9bde5" title="Click to view the MathML source">cclass="mathContainer hidden">class="mathCode">$c$. We prove that the flow will preserve the convexity for any class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003211&_mathId=si1.gif&_user=111111111&_pii=S0362546X15003211&_rdoc=1&_issn=0362546X&md5=4ff60d1bb3c575bb13422cd75ef9bde5" title="Click to view the MathML source">cclass="mathContainer hidden">class="mathCode">$c$. When class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003211&_mathId=si3.gif&_user=111111111&_pii=S0362546X15003211&_rdoc=1&_issn=0362546X&md5=0e56a77281a429a18893fd8ec475cb1b" title="Click to view the MathML source">c<gclass="mathContainer hidden">class="mathCode">$c<g$ on the initial surface, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003211&_mathId=si4.gif&_user=111111111&_pii=S0362546X15003211&_rdoc=1&_issn=0362546X&md5=c69299fbfcff6fe92c9a754d411232c8" title="Click to view the MathML source">gclass="mathContainer hidden">class="mathCode">$g$ is the fully nonlinear function, we prove that the flow will expand the hypersurface for all time. If on initial surface class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003211&_mathId=si5.gif&_user=111111111&_pii=S0362546X15003211&_rdoc=1&_issn=0362546X&md5=03be26375192506cd817cb1b70297ea0" title="Click to view the MathML source">M₀class="mathContainer hidden">class="mathCode">$M_0$ the minimal principal radius of curvature class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003211&_mathId=si6.gif&_user=111111111&_pii=S0362546X15003211&_rdoc=1&_issn=0362546X&md5=978d98ad62bde646e8ce7d1e97219deb">class="imgLazyJSB inlineImage" height="8" width="24" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0362546X15003211-si6.gif">class="mathContainer hidden">class="mathCode">$r_{\min }$ satisfies class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003211&_mathId=si7.gif&_user=111111111&_pii=S0362546X15003211&_rdoc=1&_issn=0362546X&md5=25dea47df8c9044d7a146152aeb1e232">class="imgLazyJSB inlineImage" height="9" width="58" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0362546X15003211-si7.gif">class="mathContainer hidden">class="mathCode">$r_{\min }>nc$ and minimum of the support function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003211&_mathId=si8.gif&_user=111111111&_pii=S0362546X15003211&_rdoc=1&_issn=0362546X&md5=050bece86292fb8b8c475bb26dda32d7" title="Click to view the MathML source">sclass="mathContainer hidden">class="mathCode">$s$ satisfies class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003211&_mathId=si9.gif&_user=111111111&_pii=S0362546X15003211&_rdoc=1&_issn=0362546X&md5=f7ef37c02fe70921fcf07fd657c7f9c8">class="imgLazyJSB inlineImage" height="9" width="57" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0362546X15003211-si9.gif">class="mathContainer hidden">class="mathCode">$s_{\min }>nc$, then after a scaling the hypersurface will converge to a sphere. If class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003211&_mathId=si10.gif&_user=111111111&_pii=S0362546X15003211&_rdoc=1&_issn=0362546X&md5=97683110ebb4418a45acc95b3d98dbb4" title="Click to view the MathML source">c>gclass="mathContainer hidden">class="mathCode">$c>g$ on the initial surface class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003211&_mathId=si5.gif&_user=111111111&_pii=S0362546X15003211&_rdoc=1&_issn=0362546X&md5=03be26375192506cd817cb1b70297ea0" title="Click to view the MathML source">M₀class="mathContainer hidden">class="mathCode">$M_0$ and the diameter of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003211&_mathId=si5.gif&_user=111111111&_pii=S0362546X15003211&_rdoc=1&_issn=0362546X&md5=03be26375192506cd817cb1b70297ea0" title="Click to view the MathML source">M₀class="mathContainer hidden">class="mathCode">$M_0$ satisfies class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15003211&_mathId=si13.gif&_user=111111111&_pii=S0362546X15003211&_rdoc=1&_issn=0362546X&md5=e3e1c7e8a5f0ba1075643134fc077889">class="imgLazyJSB inlineImage" height="15" width="109" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0362546X15003211-si13.gif">class="mathContainer hidden">class="mathCode">$\mathrm{diam}\left(M_0\right)<\sqrt{2}nc$, we show that the maximal existence time of the flow is finite.