Analytic and geometric problems related to capillary hypersurfaces
Abstract: This thesis consists of two parts. The first part is devoted to studying the Alexandrov−Fenchel inequalities for capillary hypersurfaces supported on geodesic planes in Euclidean space $ℝⁿ⁺¹$ and hyperbolic space $ℍⁿ⁺¹$. we introduce a locally constrained mean curvature flow in $ℝⁿ⁺¹$ and a locally constrained inverse curvature flow in $ℍⁿ⁺¹$ respectively. Along curvature flows, we show that the quermassintegrals for the capillary hypersurfaces satisfy some monotone properties and we prove the long−time existence and convergence of those flows. The second part of this thesis concerns the Minkowski problem for convex capillary hypersurfaces in $̄{ℝ}ⁿ⁺¹₊$. Namely, we want to find a convex capillary hypersurface $ ⊂̄{ℝ}ⁿ⁺¹₊$ whose Gauss−Kronecker curvature is prescribed as a positive function $f$ defined on $ _\ₜₕₑₜₐ$. We transform the capillary Minkowski problem into solving a Monge−Amp\`ere equation on spherical cap $ _\ₜₕₑₜₐ$ with a Robin boundary value condition
- Standort
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Deutsche Nationalbibliothek Frankfurt am Main
- Umfang
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Online-Ressource
- Sprache
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Englisch
- Anmerkungen
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Universität Freiburg, Dissertation, 2024
- Schlagwort
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Hyperfläche
Differentialgeometrie
- Ereignis
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Veröffentlichung
- (wo)
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Freiburg
- (wer)
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Universität
- (wann)
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2024
- Urheber
- Beteiligte Personen und Organisationen
- DOI
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10.6094/UNIFR/256604
- URN
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urn:nbn:de:bsz:25-freidok-2566046
- Rechteinformation
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Open Access; Der Zugriff auf das Objekt ist unbeschränkt möglich.
- Letzte Aktualisierung
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25.03.2025, 13:47 MEZ
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- 2024
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