A notion of the weighted σ_k-curvature for manifolds with density

详细信息    查看全文

• 作者：Jeffrey S. Case¹ ; ^{jscase@math.princeton.edu" class="auth_mail" title="E-mail the corresponding author} ; ^{jscase@psu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 53C21 ; secondary ; 35J60 ; 58E11
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：4 June 2016
• 年：2016
• 卷：295
• 期：Complete
• 页码：150-194
• 全文大小：732 K

文摘

We propose a natural definition of the weighted class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816001146&_mathId=si1.gif&_user=111111111&_pii=S0001870816001146&_rdoc=1&_issn=00018708&md5=c8914d0ee6aabd83daa82e2a17099f64" title="Click to view the MathML source">σ_kclass="mathContainer hidden">class="mathCode">${\sigma }_k$-curvature for a manifold with density; i.e. a triple class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816001146&_mathId=si2.gif&_user=111111111&_pii=S0001870816001146&_rdoc=1&_issn=00018708&md5=a8e4dab76ac83b307e23650ba18d32f4" title="Click to view the MathML source">(Mⁿ,g,e^−ϕdvol)class="mathContainer hidden">class="mathCode">$(M^n,g,e^{-\varphi }\mathrm{dvol})$. This definition is intended to capture the key properties of the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816001146&_mathId=si1.gif&_user=111111111&_pii=S0001870816001146&_rdoc=1&_issn=00018708&md5=c8914d0ee6aabd83daa82e2a17099f64" title="Click to view the MathML source">σ_kclass="mathContainer hidden">class="mathCode">${\sigma }_k$-curvatures in conformal geometry with the role of pointwise conformal changes of the metric replaced by pointwise changes of the measure. We justify our definition through three main results. First, we show that shrinking gradient Ricci solitons are local extrema of the total weighted class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816001146&_mathId=si1.gif&_user=111111111&_pii=S0001870816001146&_rdoc=1&_issn=00018708&md5=c8914d0ee6aabd83daa82e2a17099f64" title="Click to view the MathML source">σ_kclass="mathContainer hidden">class="mathCode">${\sigma }_k$-curvature functionals when the weighted class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816001146&_mathId=si1.gif&_user=111111111&_pii=S0001870816001146&_rdoc=1&_issn=00018708&md5=c8914d0ee6aabd83daa82e2a17099f64" title="Click to view the MathML source">σ_kclass="mathContainer hidden">class="mathCode">${\sigma }_k$-curvature is variational. Second, we characterize the shrinking Gaussians as measures on Euclidean space in terms of the total weighted class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816001146&_mathId=si1.gif&_user=111111111&_pii=S0001870816001146&_rdoc=1&_issn=00018708&md5=c8914d0ee6aabd83daa82e2a17099f64" title="Click to view the MathML source">σ_kclass="mathContainer hidden">class="mathCode">${\sigma }_k$-curvature functionals. Third, we characterize when the weighted class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816001146&_mathId=si1.gif&_user=111111111&_pii=S0001870816001146&_rdoc=1&_issn=00018708&md5=c8914d0ee6aabd83daa82e2a17099f64" title="Click to view the MathML source">σ_kclass="mathContainer hidden">class="mathCode">${\sigma }_k$-curvature is variational. These results are all analogues of their conformal counterparts, and in the case class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816001146&_mathId=si3.gif&_user=111111111&_pii=S0001870816001146&_rdoc=1&_issn=00018708&md5=c273a68337faba273c5a12ea978c04f7" title="Click to view the MathML source">k=1class="mathContainer hidden">class="mathCode">$k=1$ recover some of the well-known properties of Perelman's class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816001146&_mathId=si11.gif&_user=111111111&_pii=S0001870816001146&_rdoc=1&_issn=00018708&md5=5a81b55d50947421d85de36f447f8382" title="Click to view the MathML source">Wclass="mathContainer hidden">class="mathCode">$\mathcal{W}$-functional.