University of Illinois Urbana-Champaign

Volume and Energy Stability for Immersions (Harmonic, Minimal)

Hvidsten, Michael David

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Description

Title
Volume and Energy Stability for Immersions (Harmonic, Minimal)
Author(s)
Hvidsten, Michael David
Issue Date
1985
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
  • Much work has been done on minimal submanifolds of a given manifold and also on harmonic maps between manifolds. It is known, for instance, that the critical sets of the volume functional and the energy functional coincide for a Riemannian immersion f between two Riemannian manifolds.
  • An interesting question is whether the stability of f coincides for volume and energy. The second variation formulas for volume and energy show us that for variations of f in normal directions, energy stability implies volume stability. Thus, one suspects that there are energy stable harmonic immersions that are volume unstable minimal immersions.
  • We find a minimal (harmonic) immersion f that is energy stable but not volume stable by looking at maps f: M (--->) N where N is a flat Riemannian manifold, M is a compact manifold without boundary, and dim N = dim M + 1, dim M (GREATERTHEQ) 2. We show that f is a minimal, volume stable immersion if f is totally geodesic. On the other hand, for f: M (--->) N with f harmonic and N flat, we get that f is automatically energy stable.
  • For oriented surfaces M of genus g immersed in the torus T('3), we show that for g = 0 there are no minimal immersions of M in T('3). For g = 1, we get that M must be totally geodesic and thus a sub-torus. For g (GREATERTHEQ) 2, we show that f cannot be totally geodesic. Thus, a minimal (harmonic) surface in T('3) of genus g (GREATERTHEQ) 2 must be area unstable, but energy stable.
  • One such minimally immersed surface of genus 9 can be constructed from Schwarz's tetrahedral surface. This surface is a minimal surface immersed in R('3) that is triply periodic, but not oriented. By taking an 8-fold covering on this surface, and then dividing out by the periodic action, we get a minimal surface in T('3) of genus 9 that is orientable. This surface will then be area unstable, but energy stable. Several other examples of surfaces with this stability behavior for energy and volume are also discussed.
Graduation Semester
1985
Type of Resource
text
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http://hdl.handle.net/2142/71237

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