We provide geometric inequalities on $R^n$ and on general manifolds with nonnegative Ricci curvature by employing suitable monotone quantities along the flow of capacitary and $p$-capacitary potentials, as well as through related boundary value problems. Among the main achievements, we cite
[(i)] a Willmore-type inequality on manifolds with nonnegative Ricci curvature leading in turn to the sharp Isoperimetric Inequality on $3$-manifolds with nonnegative Ricci curvature ;
[(ii)] enhanced Kasue/Croke-Kleiner splitting theorems ;
[(iii)] a generalised Minkowski-type inequality in $R^n$ holding with no assumptions on the boundary of the domain considered except for smoothness ;
[(iv)] a complete discussion of maximal volume solutions to the least area problem with obstacle on Riemannian manifolds and its relation
with the variational $p$-capacity.
| Identifer | oai:union.ndltd.org:unitn.it/oai:iris.unitn.it:11572/252169 |
| Date | 13 February 2020 |
| Creators | Fogagnolo, Mattia |
| Contributors | Fogagnolo, Mattia, Mazzieri, Lorenzo |
| Publisher | Università degli studi di Trento, place:Trento |
| Source Sets | Università di Trento |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/doctoralThesis |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | firstpage:1, lastpage:164, numberofpages:164 |
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