Minimal surfaces in a Randers sphere with the rotational Killing vector field

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• 作者：Ningwei Cui ^{ningweicui@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 September 2016
• 年：2016
• 卷：441
• 期：1
• 页码：364-374
• 全文大小：424 K

文摘

The minimal surfaces in Finsler geometry with respect to the Busemann–Hausdorff measure and the Holmes–Thompson measure are called BH-minimal and HT-minimal surfaces, respectively. Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300506&_mathId=si1.gif&_user=111111111&_pii=S0022247X16300506&_rdoc=1&_issn=0022247X&md5=17ca7f49269a057a79ca2d7783267e61" title="Click to view the MathML source">(p¹,p²,p³,p⁴)class="mathContainer hidden">class="mathCode"> be the coordinates of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300506&_mathId=si2.gif&_user=111111111&_pii=S0022247X16300506&_rdoc=1&_issn=0022247X&md5=32193ebbaa219b8934020f6e4586ee82" title="Click to view the MathML source">R⁴class="mathContainer hidden">class="mathCode"> and class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300506&_mathId=si292.gif&_user=111111111&_pii=S0022247X16300506&_rdoc=1&_issn=0022247X&md5=00824cc64ac47f4cd030e03c7f285d99">class="imgLazyJSB inlineImage" height="19" width="50" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16300506-si292.gif">cript>cal-align:bottom" width="50" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16300506-si292.gif">cript>class="mathContainer hidden">class="mathCode"> be a Randers sphere of flag curvature class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300506&_mathId=si4.gif&_user=111111111&_pii=S0022247X16300506&_rdoc=1&_issn=0022247X&md5=01fee71a3b5566299393423c5cddf5fc" title="Click to view the MathML source">K=1class="mathContainer hidden">class="mathCode"> with the navigation data class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300506&_mathId=si5.gif&_user=111111111&_pii=S0022247X16300506&_rdoc=1&_issn=0022247X&md5=d773311683ebbb041fdb7c1e47533996">class="imgLazyJSB inlineImage" height="19" width="46" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16300506-si5.gif">cript>cal-align:bottom" width="46" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16300506-si5.gif">cript>class="mathContainer hidden">class="mathCode">, where class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300506&_mathId=si293.gif&_user=111111111&_pii=S0022247X16300506&_rdoc=1&_issn=0022247X&md5=0c0d0b996b44ab2c01d9523e31ba7929">class="imgLazyJSB inlineImage" height="15" width="12" 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-S0022247X16300506-si293.gif">cript>cal-align:bottom" width="12" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16300506-si293.gif">cript>class="mathContainer hidden">class="mathCode"> is the standard sphere metric and class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300506&_mathId=si115.gif&_user=111111111&_pii=S0022247X16300506&_rdoc=1&_issn=0022247X&md5=b65de1706b8d732754e06a8ae81490bb">class="imgLazyJSB inlineImage" height="19" width="141" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16300506-si115.gif">cript>cal-align:bottom" width="141" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16300506-si115.gif">cript>class="mathContainer hidden">class="mathCode">, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300506&_mathId=si295.gif&_user=111111111&_pii=S0022247X16300506&_rdoc=1&_issn=0022247X&md5=8ab37e9d5e497af3eb11e1a742a0d464" title="Click to view the MathML source">0<ε<1class="mathContainer hidden">class="mathCode">, is a Killing vector field. In this paper, we study the rotationally invariant minimal surface in class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300506&_mathId=si292.gif&_user=111111111&_pii=S0022247X16300506&_rdoc=1&_issn=0022247X&md5=00824cc64ac47f4cd030e03c7f285d99">class="imgLazyJSB inlineImage" height="19" width="50" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16300506-si292.gif">cript>cal-align:bottom" width="50" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16300506-si292.gif">cript>class="mathContainer hidden">class="mathCode"> generated by rotating the curve class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300506&_mathId=si9.gif&_user=111111111&_pii=S0022247X16300506&_rdoc=1&_issn=0022247X&md5=cef3b639ebbd4b914b5c91ee1c2e1e53" title="Click to view the MathML source">(x(s),y(s),z(s),0)class="mathContainer hidden">class="mathCode"> in the upper half sphere of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300506&_mathId=si10.gif&_user=111111111&_pii=S0022247X16300506&_rdoc=1&_issn=0022247X&md5=9f1bfa289b9fdf9b56d7dbb209b81e1e" title="Click to view the MathML source">S²class="mathContainer hidden">class="mathCode"> around the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300506&_mathId=si11.gif&_user=111111111&_pii=S0022247X16300506&_rdoc=1&_issn=0022247X&md5=f9b9b28d70683b5132dd2738d2441dfa" title="Click to view the MathML source">p¹p²class="mathContainer hidden">class="mathCode">-plane, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300506&_mathId=si12.gif&_user=111111111&_pii=S0022247X16300506&_rdoc=1&_issn=0022247X&md5=7af0d8f9caa756d009ae56c1d355d420" title="Click to view the MathML source">s∈Rclass="mathContainer hidden">class="mathCode">. We first show that such a rotational BH-minimal surface in class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300506&_mathId=si292.gif&_user=111111111&_pii=S0022247X16300506&_rdoc=1&_issn=0022247X&md5=00824cc64ac47f4cd030e03c7f285d99">class="imgLazyJSB inlineImage" height="19" width="50" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16300506-si292.gif">cript>cal-align:bottom" width="50" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16300506-si292.gif">cript>class="mathContainer hidden">class="mathCode"> is either a great 2-sphere or the catenoid in class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300506&_mathId=si13.gif&_user=111111111&_pii=S0022247X16300506&_rdoc=1&_issn=0022247X&md5=0b5523cace8e278bf2324b2f08f20508">class="imgLazyJSB inlineImage" height="19" width="47" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16300506-si13.gif">cript>cal-align:bottom" width="47" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16300506-si13.gif">cript>class="mathContainer hidden">class="mathCode">. Then we give a classification of the rotational HT-minimal surfaces, where we use the angle data to analyze the solutions of the system of ODE that characterizes the HT-minimality and prove that, such a rotational HT-minimal surface must be a great 2-sphere, an HT-minimal torus, or a rotational surface of unduloid type. As a special case, we obtain a distinguished embedded compact HT-minimal torus depending on ε. The completeness of these surfaces is also studied.