On diameter controls and smooth convergence away from singularities

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• 作者：Sajjad Lakzian¹ ; ^{slakzian@fordham.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：53C20 ; 51Fxx
• 刊名：Differential Geometry and its Applications
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：47
• 期：Complete
• 页码：99-129
• 全文大小：605 K

文摘

We prove that if a family of metrics, an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si1.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=66a8ff8c717d2acfd2bf9527f27d6a82" title="Click to view the MathML source">g_ian>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">giath>an>an>an>, on a compact Riemannian manifold, an id="mmlsi139" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si139.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=04fced88187a7b7f7613be3c5480ac55" title="Click to view the MathML source">Mⁿan>an class="mathContainer hidden">an class="mathCode">ath altimg="si139.gif" overflow="scroll">Mnath>an>an>an>, have a uniform lower Ricci curvature bound and converge to an id="mmlsi3" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si3.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=f5c616a95cb6a6df2330c31a4d4a14a7" title="Click to view the MathML source">g_∞an>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">g∞ath>an>an>an> smoothly away from a singular set, S  , with Hausdorff measure, an id="mmlsi213" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si213.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=51caedd05cd403918d84e3ba3af503e3" title="Click to view the MathML source">Hⁿ⁻¹(S)=0an>an class="mathContainer hidden">an class="mathCode">ath altimg="si213.gif" overflow="scroll">Hn−1alse">(Salse">)=0ath>an>an>an>, and if there exists connected precompact exhaustion, an id="mmlsi29" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si29.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=79b08e0191500b66415abb8d0a8ae26b" title="Click to view the MathML source">W_jan>an class="mathContainer hidden">an class="mathCode">ath altimg="si29.gif" overflow="scroll">Wjath>an>an>an>, of an id="mmlsi215" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si215.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=89f77bbbde9bb9a416f603cb6fab0603" title="Click to view the MathML source">Mⁿ∖San>an class="mathContainer hidden">an class="mathCode">ath altimg="si215.gif" overflow="scroll">Mn∖Sath>an>an>an> satisfying an id="mmlsi7" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si7.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=9dd807345ddf1f76fa3f06563dd16615" title="Click to view the MathML source">diam_gi(Mⁿ)≤D₀an>an class="mathContainer hidden">an class="mathCode">ath altimg="si7.gif" overflow="scroll">athvariant="normal">diamgialse">(Mnalse">)≤D0ath>an>an>an>, an id="mmlsi8" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si8.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=a93bad83ebceaca80e59e996b6911c17" title="Click to view the MathML source">Vol_gi(&part;W_j)≤A₀an>an class="mathContainer hidden">an class="mathCode">ath altimg="si8.gif" overflow="scroll">athvariant="normal">Volgialse">(&part;Wjalse">)≤A0ath>an>an>an> and an id="mmlsi9" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si9.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=aace24d587fb2917a7da76369e274715" title="Click to view the MathML source">Vol_gi(Mⁿ∖W_j)≤V_jan>an class="mathContainer hidden">an class="mathCode">ath altimg="si9.gif" overflow="scroll">athvariant="normal">Volgialse">(Mn∖Wjalse">)≤Vjath>an>an>an> where an id="mmlsi10" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si10.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=cf530120df67e09b8a092dd80a4531e0" title="Click to view the MathML source">lim_j&rarr;∞⁡V_j=0an>an class="mathContainer hidden">an class="mathCode">ath altimg="si10.gif" overflow="scroll">athvariant="normal">limjalse">&rarr;∞⁡Vj=0ath>an>an>an> then the Gromov&ndash;Hausdorff limit exists and agrees with the metric completion of an id="mmlsi11" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si11.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=0891292ba5cf9d4f0ac31d60ef563d58" title="Click to view the MathML source">(Mⁿ∖S,g_∞)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si11.gif" overflow="scroll">alse">(Mn∖S,g∞alse">)ath>an>an>an>. This is a strong improvement over prior work of the author with Sormani that had the additional assumption that the singular set had to be a smooth submanifold of codimension two. We have a second main theorem in which the Hausdorff measure condition on S   is replaced by diameter estimates on the connected components of the boundary of the exhaustion, an id="mmlsi12" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si12.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=8b2f95e02aa6579914f3907c9bbbfb6d" title="Click to view the MathML source">&part;W_jan>an class="mathContainer hidden">an class="mathCode">ath altimg="si12.gif" overflow="scroll">&part;Wjath>an>an>an>. This second theorem allows for singular sets which are open subregions of the manifold. In addition, we show that the uniform lower Ricci curvature bounds in these theorems can be replaced by the existence of a uniform linear contractibility function. If this condition is removed altogether, then we prove that an id="mmlsi13" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si13.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=aeeac018c9e8782ce385cc5c4778d2c1">ass="imgLazyJSB inlineImage" height="21" width="170" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0926224516000127-si13.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si13.gif" overflow="scroll">athvariant="normal">limjalse">&rarr;∞⁡dathvariant="script">False">(Mj′,N′alse">)=0ath>an>an>an>, in which an id="mmlsi14" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si14.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=a857a32b1de2a4246a50050b9f54082c">ass="imgLazyJSB inlineImage" height="20" width="23" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0926224516000127-si14.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si14.gif" overflow="scroll">Mj′ath>an>an>an> and an id="mmlsi15" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si15.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=787bbc905f2285320b12be7780e1ede4" title="Click to view the MathML source">N^′an>an class="mathContainer hidden">an class="mathCode">ath altimg="si15.gif" overflow="scroll">N′ath>an>an>an> are the settled completions of an id="mmlsi131" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si131.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=24cc2093258db38696f1f2e7b6c26bd5" title="Click to view the MathML source">(M,g_j)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si131.gif" overflow="scroll">alse">(M,gjalse">)ath>an>an>an> and an id="mmlsi17" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si17.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=a782e81b8067f46c4b7a4f1009af5a15" title="Click to view the MathML source">(M_∞∖S,g_∞)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si17.gif" overflow="scroll">alse">(M∞∖S,g∞alse">)ath>an>an>an> respectively and an id="mmlsi18" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000127&_mathId=si18.gif&_user=111111111&_pii=S0926224516000127&_rdoc=1&_issn=09262245&md5=a731642caf64f47bcafa5da96445e7b6" title="Click to view the MathML source">d_Fan>an class="mathContainer hidden">an class="mathCode">ath altimg="si18.gif" overflow="scroll">dathvariant="script">Fath>an>an>an> is the Sormani&ndash;Wenger Intrinsic Flat distance. We present examples demonstrating the necessity of many of the hypotheses in our theorems.