{"id":61899,"date":"2025-12-07T06:08:52","date_gmt":"2025-12-06T22:08:52","guid":{"rendered":"https:\/\/www.danran1967.com\/?p=61899"},"modified":"2025-12-07T06:08:52","modified_gmt":"2025-12-06T22:08:52","slug":"%e6%95%b0%e5%ad%a6%e9%ac%bc%e6%89%8d%e4%bd%a9%e9%9b%b7%e5%b0%94%e6%9b%bc","status":"publish","type":"post","link":"https:\/\/www.danran1967.com\/?p=61899","title":{"rendered":"\u6570\u5b66\u9b3c\u624d\u4f69\u96f7\u5c14\u66fc"},"content":{"rendered":"<p class=\"\"><span class=\"\"><em>\u4f5c\u8005: \u5357\u65b9\u5468\u672b\u8bb0\u8005 \u9ec4\u6c38\u660e\u00a0<\/em><\/span><\/p>\n<p><q>\u5373\u4fbf\u662f\u5728\u201c\u602a\u4eba\u201d\u4e91\u96c6\u7684\u6570\u5b66\u5bb6\u7fa4\u4f53\u4e2d\uff0c\u683c\u91cc\u6208\u91cc\u00b7\u4f69\u96f7\u5c14\u66fc\uff08Grigori 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src=\"https:\/\/q6.itc.cn\/q_70\/images03\/20250319\/6e1e53eb01094d77bca1d557574a2a69.png\" 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6\u505a\u4e86\u9898\u4e3a\u201c\u66f2\u7387\u6709\u4e0b\u754c\u7684Alexandrov\u7a7a\u95f4\u201d\u7684\u62a5\u544a\u3002\u8fd9\u4e2a\u9898\u76ee\u7684\u8bba\u6587\u5728\u4e00\u5e74\u540e\u53d1\u8868\uff0c\u6210\u4e3a\u4ed6\u7684\u4ee3\u8868\u4f5c\u4e4b\u4e00\u3002<\/p>\n<p>\u5728\u51e0\u4f55\u8282\u671f\u95f4\uff0c\u683c\u7f57\u83ab\u592b\u5411\u5404\u4e2a\u91cd\u8981\u7684\u4eba\u58eb\u4ecb\u7ecd\u4e86\u4f69\u96f7\u5c14\u66fc\uff0c\u4f7f\u5f97\u8fd9\u6b21\u65c5\u884c\u8ba9\u4f69\u96f7\u5c14\u66fc\u83b7\u5f97\u4e86\u5230\u7f8e\u56fd\u505a\u535a\u58eb\u540e\u5de5\u4f5c\u7684\u673a\u4f1a\u3002<\/p>\n<p>\u6770\u592b\u00b7\u9f50\u6770\uff08Jef Cheeger\uff09\u662f\u7f8e\u56fd\u7ebd\u7ea6\u5927\u5b66\u5e93\u6717\uff08Courant\uff09\u6570\u5b66\u7814\u7a76\u6240\u7684\u6570\u5b66\u5bb6\uff0c\u4ed6\u5728\u8fd9\u4e00\u5c4a\u7684\u51e0\u4f55\u8282\u4e0a\u4e5f\u6709\u62a5\u544a\u3002\u4ed6\u6ce8\u610f\u5230\u4e86\u4f69\u96f7\u5c14\u66fc\u3002\u4ed6\u5728\u683c\u7f57\u83ab\u592b\u7684\u4ecb\u7ecd\u4e4b\u4e0b\u4e0e\u4f69\u96f7\u5c14\u66fc\u4f1a\u9762\u3002\u4e00\u5e74\u4e4b\u540e\uff0c\u4e5f\u5c31\u662f1992\u5e74\u7684\u79cb\u5929\uff0c\u4f69\u96f7\u5c14\u66fc\u6765\u5230\u5e93\u6717\u7814\u7a76\u6240\uff0c\u5f00\u59cb\u4e86\u4ed6\u7684\u535a\u58eb\u540e\u65f6\u5149\u3002<\/p>\n<p>\u5373\u4fbf\u662f\u5728\u201c\u602a\u4eba\u201d\u4e91\u96c6\u7684\u6570\u5b66\u5bb6\u7fa4\u4f53\u4e2d\uff0c\u4f69\u96f7\u5c14\u66fc\u4e5f\u662f\u4e00\u4e2a\u7279\u6b8a\u7684\u602a\u4eba\u3002\u4ed6\u4f3c\u4e4e\u6c38\u8fdc\u90fd\u7a7f\u540c\u4e00\u4ef6\u8863\u670d\uff0c\u80e1\u5b50\u62c9\u78b4\uff0c\u4e0d\u526a\u6307\u7532\u2014\u2014\u4ed6\u8ba4\u4e3a\u8fd9\u6837\u624d\u662f\u6307\u7532\u7684\u81ea\u7136\u72b6\u6001\u3002\u4ed6\u7684\u98df\u7269\u53ea\u6709\u9762\u5305\u548c\u9178\u5976\u3002\u7f8e\u56fd\u7684\u9762\u5305\u5bf9\u4ed6\u6765\u8bf4\u53ef\u80fd\u5e76\u4e0d\u597d\u5403\uff0c\u597d\u5728\u4ed6\u627e\u5230\u4e86\u4e00\u5bb6\u552e\u5356\u6b63\u5b97\u4fc4\u7f57\u65af\u9762\u5305\u7684\u5546\u5e97\uff0c\u7ecf\u5e38\u6b65\u884c\u4e00\u6bb5\u8ddd\u79bb\u5230\u90a3\u91cc\u4e70\u9762\u5305\u3002\u6240\u4ee5\uff0c\u4ed6\u6ca1\u6709\u4ec0\u4e48\u5730\u65b9\u9700\u8981\u5f00\u9500\uff0c\u4ed6\u628a\u6240\u6709\u7684\u6d25\u8d34\u90fd\u7559\u5728\u94f6\u884c\u91cc\uff08\u8fd9\u4e3a\u4ed6\u5b58\u4e86\u4e00\u7b14\u94b1\uff0c\u4fdd\u8bc1\u540e\u6765\u7684\u4e00\u6bb5\u65f6\u95f4\u91cc\u4ed6\u80fd\u5728\u4fc4\u7f57\u65af\u6e29\u9971\u65e0\u5fe7\uff09\u3002<\/p>\n<p>\u4f69\u96f7\u5c14\u66fc\u4e00\u8f88\u5b50\u90fd\u6ca1\u6709\u79bb\u5f00\u8fc7\u4ed6\u7684\u6bcd\u4eb2\u3002\u5728\u7ebd\u7ea6\u505a\u535a\u58eb\u540e\u671f\u95f4\uff0c\u4ed6\u7684\u6bcd\u4eb2\u968f\u4ed6\u6765\u5230\u7f8e\u56fd\uff0c\u4f4f\u5728\u5e03\u9c81\u514b\u6797\uff0c\u7167\u987e\u4f69\u96f7\u5c14\u66fc\u7684\u65e5\u5e38\u751f\u6d3b\u3002<\/p>\n<p>\u6211\u4eec\u4e0d\u77e5\u9053\u4f69\u96f7\u5c14\u66fc\u5728\u4ed6\u7684\u4e00\u751f\u4e2d\u6709\u8fc7\u591a\u5c11\u4e2a\u670b\u53cb\uff0c\u4f46\u53ef\u4ee5\u80af\u5b9a\u7684\u662f\uff0c\u6570\u91cf\u975e\u5e38\u5c11\u3002\u5728\u7ebd\u7ea6\u5927\u5b66\u671f\u95f4\uff0c\u4ed6\u96be\u5f97\u5730\u4ea4\u5230\u4e86\u4e00\u4e2a\u670b\u53cb\u3002\u4f69\u96f7\u5c14\u66fc\u7684\u8001\u5e08\u7ef4\u514b\u6258\u00b7\u67e5\u52a0\u52d2\uff08Viktor Zalgaler\uff09\u975e\u5e38\u80af\u5b9a\u8fd9\u4e00\u70b9\u3002\u4ed6\u7684\u8fd9\u4f4d\u670b\u53cb\u5c31\u662f\u7530\u521a\uff0c\u73b0\u5728\u7684\u666e\u6797\u65af\u987f\u5927\u5b66\u548c\u5317\u4eac\u5927\u5b66\u6570\u5b66\u6559\u6388\u3002<\/p>\n<p>\u90a3\u4e2a\u65f6\u5019\uff0c\u4f69\u96f7\u5c14\u66fc\u7ecf\u5e38\u4e0e\u7530\u521a\u4ea4\u8c08\u3002\u4e0d\u8fc7\u5728\u7530\u521a\u7684\u8bb0\u5fc6\u4e2d\uff0c\u4ed6\u4eec\u7684\u8c08\u8bdd\u90fd\u662f\u5173\u4e8e\u6570\u5b66\u672c\u8eab\u7684\uff0c\u6ca1\u6709\u6d89\u53ca\u8fc7\u5176\u4ed6\u4e8b\u60c5\u3002\u4ed6\u8ba4\u4e3a\u4f69\u96f7\u5c14\u66fc\u4e5f\u8bb8\u4f1a\u8ddf\u5176\u4ed6\u67d0\u4e2a\u53cb\u5584\u7684\u4eba\u804a\u4e00\u804a\u5176\u4ed6\u8bdd\u9898\uff0c\u4f46\u5e76\u4e0d\u662f\u4ed6\u3002\u7530\u521a\u77e5\u9053\u4f69\u96f7\u5c14\u66fc\u4f1a\u53bb\u5e03\u9c81\u514b\u6797\u6865\u9644\u8fd1\u4e70\u9762\u5305\uff0c\u4f46\u7531\u4e8e\u7530\u521a\u672c\u4eba\u5e76\u4e0d\u5728\u4e4e\u5403\u8fd9\u79cd\u9762\u5305\u6216\u662f\u90a3\u79cd\uff0c\u6240\u4ee5\u4ed6\u4e5f\u5e76\u4e0d\u6e05\u695a\u4f69\u96f7\u5c14\u66fc\u559c\u7231\u7684\u9762\u5305\u7a76\u7adf\u6709\u4f55\u7279\u522b\u3002<\/p>\n<p>1993\u5e74\uff0c\u4f69\u96f7\u5c14\u66fc\u89e3\u51b3\u4e86\u6570\u5b66\u4e0a\u4e00\u4e2a\u957f\u671f\u5b58\u5728\u7684\u95ee\u9898\u2014\u2014\u201c\u7075\u9b42\u731c\u60f3\u201d\uff08Soul Conjecture\uff09\u3002\u8fd9\u662f\u4e00\u4e2a\u7531\u9f50\u6770\u548c\u53e6\u4e00\u540d\u6570\u5b66\u5bb6\u63d0\u51fa\u6765\u7684\u731c\u60f3\u3002\u5728\u4e8c\u5341\u5e74\u7684\u65f6\u95f4\u91cc\uff0c\u5df2\u7ecf\u6709\u4e00\u4e9b\u4eba\u5199\u4e86\u957f\u7bc7\u5927\u8bba\u6765\u5206\u6790\u8fd9\u4e2a\u95ee\u9898\uff0c\u4f46\u4ec5\u4ec5\u53ea\u80fd\u505a\u51fa\u90e8\u5206\u7684\u8bc1\u660e\u3002\u4f69\u96f7\u5c14\u66fc\u5219\u505a\u4e86\u4e00\u4e2a\u80fd\u591f\u8ba9\u6240\u6709\u4eba\u60ca\u8bb6\u4e0d\u5df2\u7684\u5b8c\u6574\u8bc1\u660e\u2014\u2014\u800c\u4e14\uff0c\u4ed6\u53ea\u7528\u4e86\u56db\u9875\u7eb8\uff01<\/p>\n<p><img src=\"https:\/\/img2.baidu.com\/it\/u=534243966,2759688270&amp;fm=253&amp;fmt=auto&amp;app=138&amp;f=JPEG?w=608&amp;h=455\" \/><\/p>\n<p>\u5bf9\u201c\u7075\u9b42\u731c\u60f3\u201d\u7684\u8bc1\u660e\uff0c\u4f7f\u5f97\u4f69\u96f7\u5c14\u66fc\u6210\u4e3a\u6570\u5b66\u754c\u7684\u5e74\u8f7b\u660e\u661f\u3002\u8fd9\u4e00\u5e74\uff0c\u4ed6\u624d27\u5c81\u3002\u4ed6\u5728\u540c\u4e00\u5e74\u7684\u79cb\u5929\u642c\u53bb\u4e86\u7f8e\u56fd\u897f\u6d77\u5cb8\u7684\u52a0\u5dde\u5927\u5b66\u7ee7\u7eed\u4ed6\u7684\u7814\u7a76\u5de5\u4f5c\u3002\u4f46\u662f\uff0c\u4f69\u96f7\u5c14\u66fc\u5f00\u59cb\u906d\u9047\u6570\u5b66\u4e0a\u7684\u5931\u8d25\uff0c\u8fd9\u5f88\u53ef\u80fd\u662f\u4ed6\u4eba\u751f\u4e2d\u7684\u7b2c\u4e00\u6b21\u5931\u8d25\u3002\u4ed6\u5728Alexandrov\u7a7a\u95f4\u7684\u7814\u7a76\u4e0a\u5361\u58f3\u4e86\uff0c\u505c\u6ede\u4e0d\u524d\u30021994\u5e74\u5f88\u53ef\u80fd\u662f\u4ee4\u4ed6\u5145\u6ee1\u4e86\u632b\u8d25\u611f\u7684\u4e00\u5e74\u3002\u540e\u6765\uff0c\u5c31\u6ca1\u6709\u4eba\u77e5\u9053\u4ed6\u7a76\u7adf\u5728\u7814\u7a76\u4ec0\u4e48\u4e86\uff0c\u76f4\u5230\u516b\u5e74\u4e4b\u540e\u4ed6\u7a81\u7136\u5728\u4e92\u8054\u7f51\u4e0a\u5f20\u8d34\u51fa\u5e9e\u52a0\u83b1\u731c\u60f3\u7684\u8bc1\u660e\u3002<\/p>\n<p>4<\/p>\n<p><q>\u57281990\u5e74\u4ee3\u89e3\u51b3\u4e86\u4e00\u7cfb\u5217\u8457\u540d\u95ee\u9898\u540e\uff0c\u4ed6\u5c31\u6d88\u5931\u4e86\u3002\u73b0\u5728\u4ed6\u53c8\u6d6e\u51fa\u4e86\u6c34\u9762\u3002<\/q><\/p>\n<p>2002\u5e7411\u670812\u65e5\uff0c\u7f8e\u56fd\u7ebd\u7ea6\u5dde\u7acb\u5927\u5b66\u6570\u5b66\u5bb6\u8fc8\u514b\u5c14\u00b7\u5b89\u5fb7\u68ee\uff08Michael Anderson\uff09\u7a81\u7136\u6536\u5230\u4e86\u4e00\u5c01\u6765\u81ea\u4f69\u96f7\u5c14\u66fc\u7684\u7535\u5b50\u90ae\u4ef6\u3002\u6b64\u65f6\u4f69\u96f7\u5c14\u66fc\u5df2\u7ecf\u56de\u56fd\u591a\u5e74\u3002\u4fe1\u4e2d\uff0c\u4f69\u96f7\u5c14\u66fc\u53ea\u8bf4\u4e86\u4e00\u53e5\u8bdd\uff1a\u201c\u6211\u60f3\u8bf7\u4f60\u7559\u610f\u6211\u5728ArXiv\u5f20\u8d34\u7684\u8bba\u6587math\u3002DG\uff0f0211159\u3002\u201d\u7136\u540e\u5c31\u662f\u8bba\u6587\u6458\u8981\u90e8\u5206\u7684\u590d\u5236\u3002<\/p>\n<p>\u5b89\u5fb7\u68ee\u662f\u5341\u6765\u540d\u6536\u5230\u76f8\u540c\u90ae\u4ef6\u7684\u6570\u5b66\u5bb6\u4e4b\u4e00\uff0c\u8fd9\u4e9b\u6570\u5b66\u5bb6\u90fd\u662f\u591a\u5e74\u6765\u4ece\u4e0d\u540c\u4fa7\u9762\u7814\u7a76\u5e9e\u52a0\u83b1\u731c\u60f3\u7684\u4eba\u58eb\u3002\u5b89\u5fb7\u68ee\u5728\u6536\u5230\u90ae\u4ef6\u7684\u7b2c\u4e8c\u5929\u51cc\u66685\u70b938\u5206\u53c8\u7ed9\u5176\u4ed6\u4e00\u4e9b\u6570\u5b66\u5bb6\u53d1\u4e86\u90ae\u4ef6\uff08\u770b\u8d77\u6765\u4ed6\u5f88\u53ef\u80fd\u5f7b\u591c\u9605\u8bfb\u4e86\u4f69\u96f7\u5c14\u66fc\u7684\u8bba\u6587\uff09\uff0c\u5e0c\u671b\u4ed6\u4eec\u80fd\u5e2e\u5fd9\u770b\u770b\u8fd9\u7bc7\u8bba\u6587\u7684\u53ef\u9760\u6027\u7a76\u7adf\u6709\u591a\u5927\u3002\u201c\u5728\u6211\u770b\u6765\u8bba\u6587\u4e2d\u7684\u60f3\u6cd5\u662f\u5168\u65b0\u7684\u548c\u539f\u521b\u7684\u2014\u2014\u5178\u578b\u7684\u683c\u91cc\u6c99\uff08\u4f69\u96f7\u5c14\u66fc\u7684\u6635\u79f0\uff09\u98ce\u683c\u3002\u201d\u5b89\u5fb7\u68ee\u5728\u90ae\u4ef6\u4e2d\u5199\u9053\u3002\u4ed6\u8fd8\u8bf4\uff1a\u201c\u4ed6\u57281990\u5e74\u4ee3\u89e3\u51b3\u4e86\u4e00\u7cfb\u5217\u5176\u4ed6\u9886\u57df\u4e2d\u8457\u540d\u7684\u95ee\u9898\uff0c\u7136\u540e\u5c31\u2018\u6d88\u5931\u2019\u4e86\u3002\u73b0\u5728\u770b\u6765\u4ed6\u53c8\u6d6e\u51fa\u4e86\u6c34\u9762\u3002\u201d<\/p>\n<p>ArXiv\u662f\u7f8e\u56fd\u5eb7\u5948\u5c14\u5927\u5b66\u56fe\u4e66\u9986\u529e\u7684\u4e00\u4e2a\u7f51\u7ad9\uff0c\u4f9b\u6570\u7406\u79d1\u5b66\u5bb6\u5f20\u8d34\u8bba\u6587\u9884\u5370\u672c\u3002\u4f69\u96f7\u5c14\u66fc\u5f20\u8d34\u7684\u8fd9\u7bc7\u8bba\u6587\u662f\u4ed6\u8bc1\u660e\u5e9e\u52a0\u83b1\u731c\u60f3\u7684\u4e09\u7bc7\u6587\u7ae0\u7684\u7b2c\u4e00\u7bc7\u3002\u7b2c\u4e8c\u7bc7\u548c\u7b2c\u4e09\u7bc7\u8bba\u6587\u57282003\u5e74\u5f20\u8d34\u3002\u6574\u4e2a\u8fc7\u7a0b\u5982\u540c\u884c\u4e91\u6d41\u6c34\uff0c\u7136\u800c\uff0c\u4ed6\u7684\u540c\u884c\u4eec\u9700\u8981\u7528\u4e00\u4e24\u5e74\u7684\u65f6\u95f4\u624d\u80fd\u7406\u89e3\u8fd9\u4e09\u7bc7\u6587\u7ae0\u3002<\/p>\n<p>2003\u5e744\u6708\uff0c\u4f69\u96f7\u5c14\u66fc\u6765\u5230\u7f8e\u56fd\u9ebb\u7701\u7406\u5de5\u5b66\u9662\uff0c\u5f00\u59cb\u4ed6\u5728\u7f8e\u56fd\u5927\u5b66\u4e2d\u7684\u5de1\u56de\u8bb2\u5ea7\u3002\u5373\u4fbf\u662f\u4ed6\u8fd9\u6837\u6c89\u9759\u3001\u5185\u5411\u3001\u4f4e\u8c03\u7684\u6570\u5b66\u5bb6\uff0c\u4e5f\u6309\u637a\u4e0d\u4f4f\u6025\u5207\u5730\u4e0e\u4eba\u5206\u4eab\u7684\u5fc3\u60c5\uff0c\u6bcf\u5929\u90fd\u5728\u7814\u8ba8\u4f1a\u4e0a\u5411\u4e0d\u540c\u7684\u542c\u4f17\u8bb2\u89e3\u4ed6\u7684\u8bc1\u660e\u3002\u4f69\u96f7\u5c14\u66fc\u975e\u5e38\u6709\u8010\u5fc3\u5730\u4e00\u70b9\u70b9\u8bb2\u89e3\uff0c\u5e76\u4e50\u4e8e\u56de\u7b54\u542c\u4f17\u63d0\u51fa\u7684\u6bcf\u4e00\u4e2a\u95ee\u9898\u3002\u5f53\u7136\uff0c\u8fd9\u79cd\u5206\u4eab\u4ec5\u9650\u4e8e\u6570\u5b66\u5708\u4e4b\u5185\uff0c\u4ed6\u53ea\u60f3\u8bb2\u7ed9\u90a3\u4e9b\u6709\u53ef\u80fd\u7406\u89e3\u4ed6\u7684\u5de5\u4f5c\u7684\u4eba\u542c\u3002<\/p>\n<p><img src=\"https:\/\/img1.baidu.com\/it\/u=3401672720,1713683821&amp;fm=253&amp;fmt=auto?w=608&amp;h=415\" 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