{"id":2263,"date":"2012-07-16T22:08:56","date_gmt":"2012-07-16T14:08:56","guid":{"rendered":"http:\/\/wamath.sinaapp.com\/?p=2263"},"modified":"2012-07-16T22:08:56","modified_gmt":"2012-07-16T14:08:56","slug":"fernando-coda-marques%e4%b8%8eandre-neves-%e8%a7%a3%e5%86%b3willmore%e7%8c%9c%e6%83%b3-2","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=2263","title":{"rendered":"Fernando Coda Marques\u4e0eAndre Neves \u89e3\u51b3Willmore\u731c\u60f3"},"content":{"rendered":"

\u7533\u660e: \u672c\u6587\u8f6c\u81eaMatheus\u2019 Weblog<\/a>, \u539f\u6587\u9898\u76ee\u4e3aThe Willmore conjecture after Fernando Coda Marques and Andre Neves<\/a><\/em>, \u539f\u6587\u4f5c\u8005:matheuscmss. \u5148\u7ffb\u8bd1\u81f3\u6b64, \u5e0c\u671b\u6709\u66f4\u591a\u7684\u4eba\u611f\u5174\u8da3.<\/p>\n

—-\u6b63\u6587\u5f00\u59cb—-<\/p>\n

\u5728\u8fc7\u53bb\u76845\u4e2a\u6708\u4e2d, \u6211\u5f88\u9ad8\u5174\u5730\u770b\u5230\u6709\u4eba\u5ba3\u79f0\u89e3\u51b3\u4e86\u6570\u5b66\u51e0\u4e2a\u9886\u57df\u4e2d\u7684\u91cd\u8981\u7684\u95ee\u9898\u548c\u731c\u60f3.\u4f8b\u5982:<\/p>\n

    \n
  1. Ian Agol \u5ba3\u79f0\u8bc1\u660e\u4e863\u7ef4\u62d3\u6251\u4e2d\u7684 virtual Haken \u731c\u60f3 (\u53c2\u8003 D. Calegari \u535a\u5ba2<\/a>\u7684\u4e09\u7bc7\u65e5\u5fd7).<\/li>\n
  2. Fernando Cod\u00e1 Marques and Andr\u00e9 Neves<\/a> \u5ba3\u79f0\u8bc1\u660e\u4e86 Willmore \u731c\u60f3<\/a> (\u53c2\u8003 F.Morgan \u7684\u8fd9\u7bc7\u6587\u7ae0<\/a>—\u975e\u6b63\u5f0f\u7684\u9610\u8ff0\u4e86\u8fd9\u4e0047\u5e74\u4e4b\u4e45\u7684\u731c\u60f3)<\/li>\n
  3. Alex Eskin, Maxim Kontsevich and Anton Zorich<\/a> \u6700\u8fd1\u5b8c\u6210\u4e86\u5173\u4e8eKontsevich-Zorich \u95ed\u4e0a\u94fe \u7684Lyapunov\u6307\u6570\u548c\u516c\u5f0f\u7684\u8bc1\u660e (\u8be5\u516c\u5f0f\u7531M.Kontsevich\u4e8e15\u5e74\u524d\u5728\u8fd9\u7bc7\u6587\u7ae0<\/a>\u4e2d\u63d0\u51fa)<\/li>\n
  4. Alex Eskin and Maryam Mirzakhani\u6700\u8fd1\u5ba3\u79f0\u8bc1\u660e\u4e86Abel\u5fae\u5206\u6784\u6210\u7684\u6a21\u7a7a\u95f4(\u975e\u9f50\u6b21)\u4e2d\u7684\u4e00\u4e2aRatner\u578b\u5b9a\u7406<\/a>(\u4e0d\u53d8\u6d4b\u5ea6\u7684\u5206\u7c7b).<\/li>\n<\/ol>\n


    \n\u7279\u522b\u662f, \u9274\u4e8e\u6211\u5df2\u7ecf\u548cFernando Cod\u00e1 Marques, Andr\u00e9 Neves, Anton Zorich and Alex Eskin\u89c1\u8fc7\u51e0\u9762, \u5728\u6b64\u6211\u5f88\u9ad8\u5174(\u4e5f\u6709\u70b9\u5192\u5931\u5730)\u5bf9\u4ed6\u4eec\u6770\u51fa\u7684\u5de5\u4f5c\u8868\u793a\u8d5e\u7f8e! (\u4e8b\u5b9e\u4e0a, \u8be5\u6587\u9664\u4e86\u8bc1\u660e\u4e86\u4e00\u4e2a\u6f02\u4eae\u7684\u7ed3\u8bba, \u800c\u4e14\u4e5f\u662f\u7b80\u660e\u7684. \u4e0a\u8ff0\u5217\u8868\u4e2d\u7684\u7b2c\u4e00, \u7b2c\u4e09, \u7b2c\u56db\u5206\u522b\u670995, 106\u4ee5\u53ca152\u9875\u957f.:-)<\/p>\n

    \u4e0b\u9762\u6211\u5c06\u5728\u672c\u535a\u5ba2\u4e2d\u5199\u4e00\u4e9b\u65e5\u5fd7\u6765\u5bf9\u524d\u9762\u63d0\u5230\u7684\u540e\u4e09\u9879\u4f5c\u4e9b\u6ce8\u8bb0. \u66f4\u786e\u5207\u7684\u8bf4, \u4eca\u5929\u7684\u8fd9\u7bc7\u65e5\u5fd7\u5c06\u5b8c\u5168\u7528\u6765\u8bf4\u660e\u7531Fernado \u548c Andr\u00e9 (\u8fd9\u91cc\u6211\u5c06\u7528\u540d\u5b57\u6765\u79f0\u547c\u4ed6\u4eec, \u56e0\u4e3a\u6211\u8ba4\u4e3a\u4ed6\u4eec\u4f1a\u4e0d\u4ecb\u610f) \u89e3\u51b3\u7684Willomroe\u731c\u60f3. \u800c\u540e, \u5c06\u5728\u53e6(\u4e00\u7bc7?)\u65e5\u5fd7\u4e2d\u8ba8\u8bbaAlex, Anton \u548c M. Kontsevich \u5173\u4e8eKontsevich-Zorich \u95ed\u4e0a\u94fe\u7684Lyapunov \u6307\u6570\u548c\u7684\u6587\u7ae0. \u6700\u540e, \u6211\u5c06\u5b8c\u6210\u8fd9\u4e00”\u7cfb\u5217”\u65e5\u5fd7, \u5e76\u7528(\u51e0\u7bc7?)\u65e5\u5fd7\u6765\u5b8c\u6210Alex \u548c M.Mirzakhani\u8bc1\u660e\u7684Ratner-\u578b\u5b9a\u7406, \u8fd9\u4e9b\u65e5\u5fd7\u5c06\u57fa\u4e8e\u524d\u51e0\u5468Alex\u5728Luminy\/Marseille\u6388\u8bfe\u7684\u7b14\u8bb0(\u7531\u6211\u8bb0\u5f55).<\/p>\n

    \u5728\u8f6c\u5165\u6b63\u9898\u4e4b\u524d, \u5148\u8ba9\u6211\u4f5c\u4e00\u4e9b\u7533\u660e\u548c\u8bc4\u8bba. \u9996\u5148, \u5199\u4f5c\u4eca\u5929\u8fd9\u7bc7\u65e5\u5fd7\u4e4b\u6240\u4ee5\u88ab\u6781\u5927\u5730\u7b80\u5316\u4e86, \u662f\u7531\u4e8eFernado\u548cAndr\u00e9\u5728\u4ed6\u4eec\u6587\u7ae0\u7684\u7b2c\u4e8c\u8282\u5bf9\u5176\u8bc1\u660e\u601d\u60f3\u5199\u4e86\u4e00\u4e2a\u975e\u5e38\u6e05\u6670\u7684\u5927\u7eb2. \u7279\u522b\u662f, \u4eca\u5929\u6211\u5c06\u8ddf\u968f\u7740\u4ed6\u4eec\u7684\u5927\u7eb2, \u4f46\u540c\u65f6\u8bf7\u6ce8\u610f, \u6211\u53ef\u80fd\u5728\u8fd9\u4e2a\u8fc7\u7a0b\u4e2d\u72af\u4e00\u4e9b\u9519\u8bef (\u5f53\u7136, \u8fd9\u6837\u7684\u9519\u8bef\u5e94\u8be5\u7531\u6211\u4e00\u4eba\u627f\u62c5). \u7b2c\u4e8c, \u7b2c\u4e09\u6761\u7684\u6587\u7ae0\u5c06\u662fPascal Hubert\u4e8e2012\u5e7410\u6708\u5728\u5e03\u5c14\u5df4\u57fa\u7814\u8ba8\u73ed(S\u00e9minaire Bourbaki)\u7684\u62a5\u544a\u4e3b\u9898. \u7279\u522b\u662f, \u4ed6\u5c06\u4e3a\u6b64\u5199\u4e00\u4e2a\u4e25\u683c\u7684\u8bb2\u4e49, \u4e8e\u662f\u6211\u5173\u4e8e\u8fd9\u4e2a\u4e3b\u9898\u7684\u65e5\u5fd7\u5e94\u5f53\u88ab\u89c6\u4e3aPascal\u7b14\u8bb0\u7684\u975e\u6b63\u5f0f\u7248\u672c. \u6700\u540e, \u5173\u4e8e\u4e0a\u9762\u63d0\u5230\u7684\u7b2c\u56db\u6761, Alex Eskin\u5728\u4ed6\u7684\u4e3b\u9875(\u53c2\u8003\u8fd9\u91cc<\/a>)\u6302\u4e86\u4e00\u4e2a\u975e\u5e38\u597d\u7684\u7b14\u8bb0(\u5173\u4e8e\u4ed6\u5f00\u8bbe\u7684\u5c0f\u8bfe\u7a0b(\u4e0d\u5230\u4e00\u4e2a\u5b66\u671f)\u7684). \u4e8b\u5b9e\u4e0a, Alex\u7684\u7b14\u8bb0\u8bfb\u8d77\u6765\u4e5f\u662f\u4ee4\u4eba\u6109\u60a6\u7684, \u4ece\u800c\u6211\u7acb\u9a6c\u6539\u53d8\u4e86\u4e00\u5f00\u59cb\u628a\u6211\u539f\u59cb\u7684\u7b14\u8bb0\u5199\u5230\u8fd9\u91cc\u7684\u521d\u8877, \u8fd9\u6837\u6211\u5c06\u505a\u4e9bAlex\u548cM. Mirzakhani\u7528\u5230\u7684\u6280\u5de7\u7684\u8bc4\u8bba, \u4f8b\u5982, \u6211\u5c06\u8bd5\u7740\u8d34\u51fa\u6211\u4eceJ.-F. Quint\u5f00\u8bbe\u7684\u5c0f\u8bfe\u7a0b(\u4e5f\u8bf7\u53c2\u8003 Luminy\/Marseille)\u8bb0\u7684\u7b14\u8bb0\u4e2d\u6240\u8c13\u7684”\u6307\u6570\u6f02\u79fb”(exponential drift)\u60f3\u6cd5(\u6765\u6e90\u4e8e\u4ed6\u548cY. Benoist\u7684\u8457\u540d\u6587\u7ae0<\/a>).<\/p>\n

    \u5728\u8bf4\u5b8c\u8fd9\u4e9b\u8bc4\u8bba\u4e4b\u540e, \u8ba9\u6211\u4eec(\u4ece\u4e0b\u6bb5)\u5f00\u59cb\u5bf9Willmore\u731c\u60f3\u4ee5\u53caFernado\u548cAndr\u00e9\u5bf9\u5176\u89e3\u51b3\u7684\u8ba8\u8bba.<\/p>\n

    \u5f15\u8a00<\/h4>\n

    \u5047\u8bbe $\\widetilde{\\Sigma}\\hookrightarrow\\mathbb{R}^3$ \u662f ${\\mathbb{R}^3}$ \u4e00\u4e2a\u95ed\u7684(\u5373\u7d27\u81f4\u4e14\u65e0\u8fb9\u754c)\u6d78\u5165\u66f2\u9762. \u5728 ${\\widetilde{\\Sigma}}$ \u7684\u6240\u6709\u6bd4\u8f83\u7b80\u5355\u7684\u51e0\u4f55\u4e0d\u53d8\u91cf\u4e2d, \u6211\u4eec\u53ef\u4ee5\u9009\u53d6(\u6bcf\u70b9 ${p\\in\\widetilde{\\Sigma}}$ \u7684)\u9ad8\u65af\u66f2\u7387 ${\\widetilde{K}=\\widetilde{K}(p)}$ \u4ee5\u53ca\u5e73\u5747\u66f2\u7387 ${\\widetilde{H}=\\widetilde{H}(p)}$, \u4ed6\u4eec\u662f\u6240\u8c13\u7684\u7b2c\u4e8c\u57fa\u672c\u578b ${\\widetilde{A}=\\widetilde{A}(p):T_p\\widetilde{\\Sigma}\\rightarrow T_p\\widetilde{\\Sigma}}$ \u7684\u884c\u5217\u5f0f(determinant)\u548c\u8ff9(trace) (\u4e00\u4e2a ${2\\times2}$ \u7684\u7ebf\u6027\u53d8\u6362\/\u77e9\u9635\u7684\u7279\u5f81\u503c\u662f ${\\widetilde{\\Sigma}}$ \u5728 $p$ \u70b9\u7684\u4e3b\u66f2\u7387(principal curvatures)). \u5173\u4e8e\u8fd9\u4e9b\u6982\u5ff5\u7684\u9610\u8ff0, \u4e00\u672c\u975e\u5e38\u597d\u7684\u53c2\u8003\u4e66\u662f M. do Carmo \u7684\u7ecf\u5178\u8457\u4f5c: \u66f2\u7ebf\u4e0e\u66f2\u9762\u7684\u5fae\u5206\u51e0\u4f55(Differential geometry of curves and surfaces).
    \n\u7528\u8fd9\u79cd(\u5fae\u5206\u51e0\u4f55\u7684)\u8bed\u8a00, Willmore \u80fd\u91cf(Willmore Energy) ${\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})}$ \u5b9a\u4e49\u4e3a
    \n\\[
    \n\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma}) = \\int_{\\widetilde{\\Sigma}} \\widetilde{H}^2\\,d\\widetilde{\\Sigma}
    \n\\]
    \n\u5176\u4e2d ${d\\widetilde{\\Sigma}}$ \u662f ${\\widetilde{\\Sigma}}$ \u7684\u9762\u79ef\u5143.
    \nWillmore \u80fd\u91cf\u81ea\u7136\u7684\u51fa\u73b0\u5728\u67d0\u4e9b\u7269\u7406(\u5728\u5f39\u6027\u58f3(elastic shells))\u548c\u751f\u7269(\u7ec6\u80de\u819c(cell membranes))\u7684\u7814\u7a76\u4e2d, \u5728\u90a3\u91cc, \u6709\u65f6\u4e5f\u88ab\u79f0\u4e3a\u626d\u66f2\u80fd\u91cf(bending energy).
    \n\u5728\u6570\u5b66\u4e2d, \u6211\u4eec\u77e5\u9053\u5b83\u5728 ${\\mathbb{R}^3}$\u4e2d\u7684\u5171\u5f62\u53d8\u6362\u4e0b\u662f\u4e0d\u53d8\u7684(\u53c2\u8003 W. Blaschke), \u800c\u4e14\u5b83\u6709\u4e00\u4e2a\u975e\u5e38\u81ea\u7136\u7684\u51e0\u4f55(\u53d8\u5206)\u7ed3\u6784,\u5373\u5728\u6240\u6709\u6d78\u5165\u95ed\u66f2\u9762 ${\\widetilde{\\Sigma}\\hookrightarrow\\mathbb{R}^{3}}$ \u4e2d\u4ed6\u4f55\u65f6\u8fbe\u5230\u6700\u5c0f?
    \n\u8fd9\u91cc, \u53ef\u4ee5\u8bc1\u660e\u5bf9\u6240\u6709\u6d78\u5165\u95ed\u66f2\u9762 ${\\widetilde{\\Sigma}\\hookrightarrow\\mathbb{R}^3}$, \u6709
    \n\\[
    \n\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})\\geq 4\\pi = \\widetilde{\\mathcal{W}}(S^2(r))
    \n\\]
    \n\u5176\u4e2d, ${S^2(r)\\subset\\mathbb{R}^3}$ \u662f\u534a\u5f84\u4e3a ${r>0}$ \u7684 (??\u6807\u51c6\u7403\u9762, \u539f\u6587\u8fd9\u91cc\u7f3a\u5931). \u800c\u4e14 ${\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})= 4\\pi}$ \u5f53\u4e14\u4ec5\u5f53 ${\\widetilde{\\Sigma}}$ \u662f\u4e00\u4e2a\u7403\u9762(\u6807\u51c6\u7403\u9762, \u975e\u62d3\u6251\u7403\u9762). \u6362\u8a00\u4e4b, \u5728 ${\\mathbb{R}^3}$ \u4e2d\u7684\u6807\u51c6\u7403\u662f\u5728\u6240\u6709\u6d78\u5165\u95ed\u66f2\u9762\u4e2d\u4f7fWillmore\u80fd\u91cf\u6781\u5c0f\u8005. \u4e8b\u5b9e\u4e0a, \u8fd9\u4e00\u4e8b\u5b9e\u7684\u8bc1\u660e\u5e76\u4e0d\u56f0\u96be:
    \n\\begin{itemize}
    \n\\item \u8bb0 ${K_1\\geq k_2}$ \u4e3a\u66f2\u9762\u7684\u4e3b\u66f2\u7387, \u56de\u5fc6\u5e73\u5747\u66f2\u7387\u662f${\\widetilde{H}=(k_1+k_2)\/2}$, \u800c\u9ad8\u65af\u66f2\u7387\u662f ${\\widetilde{K}=k_1k_2}$. \u5229\u7528\u6052\u7b49\u5f0f ${(k_1+k_2)^2=(k_1-k_2)^2+4k_1k_2}$, \u6709
    \n\\[
    \n\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})=\\int_{\\widetilde{\\Sigma}} \\widetilde{H}^2\\,d\\widetilde{\\Sigma} = \\frac{1}{4}\\int_{\\widetilde{\\Sigma}} (k_1-k_2)^2\\,d\\widetilde{\\Sigma}+\\int_{\\widetilde{\\Sigma}} \\widetilde{K}\\,d\\widetilde{\\Sigma}\\geq \\int_{\\widetilde{\\Sigma}} \\widetilde{K}\\,d\\widetilde{\\Sigma}
    \n\\]
    \n\u53e6\u4e00\u65b9\u9762, \u7531\u9ad8\u65af-\u535a\u5185\u5b9a\u7406(Gauss-Bonnet theorem), ${\\int_{\\widetilde{\\Sigma}} \\widetilde{K}\\,d\\widetilde{\\Sigma}=2\\pi\\chi(\\widetilde{\\Sigma})}$. \u56e0\u6b64, \u5047\u8bbe ${\\widetilde{\\Sigma}}$ \u662f\u62d3\u6251\u7403\u9762, \u5373\u5b83\u7684\u4e8f\u683c $g=0$, \u4ece\u800c\u6b27\u62c9\u793a\u6027\u6570\u4e3a ${\\chi(\\widetilde{\\Sigma})=2-2g=2}$, \u6211\u4eec\u53ef\u4ee5\u5f97\u5230
    \n\\[
    \n\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})\\geq \\int_{\\widetilde{\\Sigma}} \\widetilde{K}\\,d\\widetilde{\\Sigma}=4\\pi
    \n\\]
    \n\u800c\u4e14\u7b49\u53f7\u6210\u7acb\u5f53\u4e14\u4ec5\u5f53
    \n\\[
    \n\\frac{1}{4}\\int_{\\widetilde{\\Sigma}} (k_1-k_2)^2\\,d\\widetilde{\\Sigma}=0
    \n\\]
    \n\u5373 ${k_1=k_2}$ \u5bf9 ${\\widetilde{\\Sigma}}$ \u4e0a\u6240\u6709\u70b9\u90fd\u6210\u7acb. \u5373, \u6240\u6709\u70b9\u90fd\u662f\u5e73\u70b9, \u8fd9\u6837\u5c31\u8bf4\u660e ${\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})=4\\pi}$ \u6210\u7acb\u5f53\u4e14\u4ec5\u5f53 ${\\widetilde{\\Sigma}}$ \u662f\u4e2a\u6807\u51c6\u7403.
    \n\\item \u4e00\u822c\u5730(\u5373, \u5f53 ${\\widetilde{\\Sigma}}$ \u7684\u4e8f\u683c $g\\geq1$ \u65f6), \u524d\u9762\u7684\u8ba1\u7b97\u4e0d\u518d\u6709\u6548, \u56e0\u4e3a ${\\int_{\\widetilde{\\Sigma}} \\widetilde{K}\\,d\\widetilde{\\Sigma}=2\\pi\\chi(\\widetilde{\\Sigma})=2\\pi(2-2g)\\leq 0}$, \u5373\u4f9d\u8d56\u4e8e\u9ad8\u65af-\u535a\u5185\u5b9a\u7406\u800c\u5f97\u5230 ${\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})\\geq \\int_{\\widetilde{\\Sigma}} \\widetilde{K}\\,d\\widetilde{\\Sigma}}$ \u7684\u4f30\u8ba1, \u6bd4\u5e73\u51e1\u7684\u754c ${\\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})= \\int_{\\widetilde{\\Sigma}} H^2\\,d\\widetilde{\\Sigma}\\geq0}$ \u8981\u5f31. \u7136\u800c\u6211\u4eec\u53ef\u4ee5\u53d6\u800c\u4ee3\u4e4b\u7528 Chern-Lashof \u4e0d\u7b49\u5f0f
    \n\\[
    \n\\int_{\\widetilde{\\Sigma}} |\\widetilde{K}|\\,d\\widetilde{\\Sigma}\\geq 2\\pi(2+2g)
    \n\\]
    \n\u6765\u514b\u670d\u8fd9\u70b9: \u4e8b\u5b9e\u4e0a, \u82e5\u8bb0 ${a:=\\int_{\\{\\widetilde{K}>0\\}} \\widetilde{K}\\,d\\widetilde{\\Sigma}}$ \u548c ${b:=\\int_{\\{\\widetilde{K}\\leq0\\}} \\widetilde{K}\\,d\\widetilde{\\Sigma}}$, \u5219\u6211\u4eec\u6709
    \n\\[a+b=\\int_{\\widetilde{\\Sigma}} \\widetilde{K}\\,d\\widetilde{\\Sigma}=(2-2g)2\\pi
    \n\\]
    \n\u4ee5\u53ca
    \n\\[
    \na-b=\\int_{\\widetilde{\\Sigma}} |\\widetilde{K}|\\,d\\widetilde{\\Sigma}\\geq 2\\pi(2+2g)
    \n\\]
    \n\u4e8e\u662f ${a\\geq 4\\pi}$. \u73b0\u5728, \u6211\u4eec\u5bf9\u533a\u57df ${\\{\\widetilde{K}>0\\}}$ \u53ef\u4ee5\u7528 ${(k_1+k_2)^2=(k_1-k_2)^2+4k_1k_2}$ \u5f97\u5230
    \n\\[
    \n4\\pi\\leq a\\leq \\int_{\\{\\widetilde{K}>0\\}} \\widetilde{K}\\,d\\widetilde{\\Sigma} + \\frac{1}{4}\\int_{\\{\\widetilde{K}>0\\}} (k_1-k_2)^2\\,d\\widetilde{\\Sigma}\\leq \\widetilde{\\mathcal{W}}(\\widetilde{\\Sigma})
    \n\\]
    \n\\end{itemize}
    \n\u65e0\u8bba\u5bf9\u8fd9\u4e24\u79cd\u60c5\u51b5\u4e2d\u7684\u54ea\u4e00\u79cd, \u4e00\u4f46\u610f\u8bc6\u5230\u5728\u6240\u6709\u6d78\u5165${\\mathbb{R}^3}$\u7684\u7d27\u66f2\u9762\u4e2d\u6781\u5c0f\u5316Willmore\u80fd\u91cf\u5e76\u4e0d\u56f0\u96be, \u90a3\u4e48\u6211\u4eec\u4e5f\u8bb8\u4f1a\u548cT. Willmore\u4e00\u6837\u8003\u8651\u5728\u6240\u6709\u6d78\u5165${\\mathbb{R}^3}$\u7684\u73af\u9762\u4e2d\u4f7fWillmore\u80fd\u91cf\u6700\u5c0f\u8fd9\u4e2a\u95ee\u9898(\u6700\u5c0f\u662f\u591a\u5c11? \u4ec0\u4e48\u65f6\u5019\u8fbe\u5230\u8fd9\u4e2a\u6700\u5c0f?).
    \n\\begin{thm}[Willmore \u731c\u60f3, (1965)]
    \n\u5bf9\u6240\u6709\u7684\u6d78\u5165\u73af\u9762${\\widetilde{\\Sigma}\\hookrightarrow\\mathbb{R}^3}$, \u90fd\u6709
    \n\\[{\\widetilde{W}(\\widetilde{\\Sigma})\\geq 2\\pi^2(>4\\pi)}.\\]
    \n\u5bf9\u65cb\u8f6c\u73af\u9762 ${\\widetilde{\\Sigma}_c}$ \u7b49\u5f0f ${\\widetilde{W}(\\widetilde{\\Sigma}_c)=2\\pi^2}$ \u6210\u7acb, \u8fd9\u4e2a\u73af\u9762\u7531\u4e00\u4e2a\u534a\u5f84\u4e3a $1$, \u5176\u4e2d\u5fc3\u79bb\u8f74\u5411(z\u8f74)\u8ddd\u79bb\u4e3a ${\\sqrt{2}}$ \u7684\u5706\u7ed5 z\u8f74 \u65cb\u8f6c\u800c\u5f97. \u5176\u65b9\u7a0b\u53ef\u4ee5\u5199\u6210:
    \n\\[
    \n(u,v)\\mapsto ((\\sqrt{2}+\\cos u)\\cos v, (\\sqrt{2}+\\cos u) \\sin v, \\sin u)\\in\\mathbb{R}^3.
    \n\\]
    \n\\end{thm}
    \n\u5173\u4e8e\u7b49\u53f7\u53d6\u5f97\u65f6\u7684\u8fd9\u4e2a\u73af\u9762, \u53ef\u4ee5\u53c2\u8003    P.Nylander\u7684\u4e00\u7bc7\u65e5\u5fd7<\/a>, \u90a3\u91cc\u6709\u5b83\u7684\u4e00\u4e2a\u51c6\u786e\u56fe\u5f62.<\/p>\n