On Symmetric Invariants of Level Surfaces Near Regular Points

Magnanini, Rolando

doi:10.1112/blms/24.6.565

Cited by 3 publications

(1 citation statement)

References 7 publications

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“…The geometric part of the problem (which is also rather subtle, as evidenced by the examples in [13,12]) has also attracted considerable attention; in particular, an aspect that has been extensively studied is that of the curvature and shape of level sets (see e.g. [16,1,20,21,24,9] and references therein).…”

Section: Introduction and Statementsmentioning

confidence: 99%

Some geometric conjectures in harmonic function theory

Enciso

Peralta‐Salas

2011

Annali di Matematica

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We prove some results on the geometry of the level sets of harmonic functions, particularly regarding their 'oscillation' and 'pinching' properties. These results allow us to tackle three recent conjectures due to L. De Carli and S.M. Hudson (Bull. London Math. Soc. 42 (2010) 83-95). Our approach hinges on a combination of local constructions, methods from differential topology and global extension arguments.

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Section: Introduction and Statementsmentioning

confidence: 99%

Some geometric conjectures in harmonic function theory

Enciso

Peralta‐Salas

2011

Annali di Matematica

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Submanifolds that are level sets of solutions to a second-order elliptic PDE

Enciso

Peralta‐Salas

2013

Advances in Mathematics

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Abstract. Motivated by a question of Rubel, we consider the problem of characterizing which noncompact hypersurfaces in R n can be regular level sets of a harmonic function modulo a C ∞ diffeomorphism, as well as certain generalizations to other PDEs. We prove a versatile sufficient condition that shows, in particular, that any nonsingular algebraic hypersurface whose connected components are all noncompact can be transformed onto a union of components of the zero set of a harmonic function via a diffeomorphism of R n . The technique we use, which is a significant improvement of the basic strategy we recently applied to construct solutions to the Euler equation with knotted stream lines (Ann. of Math. 175 (2012) 345-367), combines robust but not explicit local constructions with appropriate global approximation theorems. In view of applications to a problem of Berry and Dennis, intersections of level sets are also studied.

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Stationary isothermic surfaces and uniformly dense domains

Magnanini¹,

Prajapat²,

Sakaguchi³

2006

Trans. Amer. Math. Soc.

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Abstract. We establish a relationship between stationary isothermic surfaces and uniformly dense domains. A stationary isothermic surface is a level surface of temperature which does not evolve with time. A domain Ω in the Ndimensional Euclidean space R N is said to be uniformly dense in a surface Γ ⊂ R N of codimension 1 if, for every small r > 0, the volume of the intersection of Ω with a ball of radius r and center x does not depend on x for x ∈ Γ.We prove that the boundary of every uniformly dense domain which is bounded (or whose complement is bounded) must be a sphere. We then examine a uniformly dense domain with unbounded boundary ∂Ω, and we show that the principal curvatures of ∂Ω satisfy certain identities.The case in which the surface Γ coincides with ∂Ω is particularly interesting. In fact, we show that, if the boundary of a uniformly dense domain is connected, then (i) if N = 2, it must be either a circle or a straight line and (ii) if N = 3, it must be either a sphere, a spherical cylinder or a minimal surface. We conclude with a discussion on uniformly dense domains whose boundary is a minimal surface.

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On Symmetric Invariants of Level Surfaces Near Regular Points

Cited by 3 publications

References 7 publications

Some geometric conjectures in harmonic function theory

Some geometric conjectures in harmonic function theory

Submanifolds that are level sets of solutions to a second-order elliptic PDE

Stationary isothermic surfaces and uniformly dense domains

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