{"title":"高维奇异旋转对称梯度利玛窦孤子的存在性","authors":"Kin Ming Hui","doi":"10.4153/s0008439524000237","DOIUrl":null,"url":null,"abstract":"<p>By using fixed point argument, we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$g=\\frac {da^2}{h(a^2)}+a^2g_{S^n}$</span></span></img></span></span> for some function <span>h</span> where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$g_{S^n}$</span></span></img></span></span> is the standard metric on the unit sphere <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$S^n$</span></span></img></span></span> in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbb {R}^n$</span></span></img></span></span> for any <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$n\\ge 2$</span></span></img></span></span>. More precisely, for any <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$\\lambda \\ge 0$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$c_0>0$</span></span></img></span></span>, we prove that there exist infinitely many solutions <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline8.png\"><span data-mathjax-type=\"texmath\"><span>${h\\in C^2((0,\\infty );\\mathbb {R}^+)}$</span></span></img></span></span> for the equation <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-\\lambda r-(n-1))$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$h(r)>0$</span></span></img></span></span>, in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline11.png\"><span data-mathjax-type=\"texmath\"><span>$(0,\\infty )$</span></span></img></span></span> satisfying <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline12.png\"><span data-mathjax-type=\"texmath\"><span>$\\underset {\\substack {r\\to 0}}{\\lim }\\,r^{\\sqrt {n}-1}h(r)=c_0$</span></span></img></span></span> and prove the higher-order asymptotic behavior of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behavior near the origin.</p>","PeriodicalId":501184,"journal":{"name":"Canadian Mathematical Bulletin","volume":"19 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions\",\"authors\":\"Kin Ming Hui\",\"doi\":\"10.4153/s0008439524000237\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>By using fixed point argument, we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline1.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$g=\\\\frac {da^2}{h(a^2)}+a^2g_{S^n}$</span></span></img></span></span> for some function <span>h</span> where <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline2.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$g_{S^n}$</span></span></img></span></span> is the standard metric on the unit sphere <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline3.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$S^n$</span></span></img></span></span> in <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline4.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathbb {R}^n$</span></span></img></span></span> for any <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline5.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$n\\\\ge 2$</span></span></img></span></span>. More precisely, for any <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline6.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\lambda \\\\ge 0$</span></span></img></span></span> and <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline7.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$c_0>0$</span></span></img></span></span>, we prove that there exist infinitely many solutions <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline8.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>${h\\\\in C^2((0,\\\\infty );\\\\mathbb {R}^+)}$</span></span></img></span></span> for the equation <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline9.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-\\\\lambda r-(n-1))$</span></span></img></span></span>, <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline10.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$h(r)>0$</span></span></img></span></span>, in <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline11.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$(0,\\\\infty )$</span></span></img></span></span> satisfying <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240403061655231-0392:S0008439524000237:S0008439524000237_inline12.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\underset {\\\\substack {r\\\\to 0}}{\\\\lim }\\\\,r^{\\\\sqrt {n}-1}h(r)=c_0$</span></span></img></span></span> and prove the higher-order asymptotic behavior of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behavior near the origin.</p>\",\"PeriodicalId\":501184,\"journal\":{\"name\":\"Canadian Mathematical Bulletin\",\"volume\":\"19 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-03-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Canadian Mathematical Bulletin\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4153/s0008439524000237\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Canadian Mathematical Bulletin","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4153/s0008439524000237","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions
By using fixed point argument, we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric $g=\frac {da^2}{h(a^2)}+a^2g_{S^n}$ for some function h where $g_{S^n}$ is the standard metric on the unit sphere $S^n$ in $\mathbb {R}^n$ for any $n\ge 2$. More precisely, for any $\lambda \ge 0$ and $c_0>0$, we prove that there exist infinitely many solutions ${h\in C^2((0,\infty );\mathbb {R}^+)}$ for the equation $2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-\lambda r-(n-1))$, $h(r)>0$, in $(0,\infty )$ satisfying $\underset {\substack {r\to 0}}{\lim }\,r^{\sqrt {n}-1}h(r)=c_0$ and prove the higher-order asymptotic behavior of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behavior near the origin.