1,775 research outputs found
Geometric Mean Curvature Lines on Surfaces Immersed in R3
Here are studied pairs of transversal foliations with singularities, defined
on the Elliptic region (where the Gaussian curvature is positive)
of an oriented surface immersed in . The leaves of the foliations
are the lines of geometric mean curvature, along which the normal curvature is
given by , which is the geometric mean curvature of the
principal curvatures of the immersion. The singularities of the
foliations are the umbilic points and parabolic curves}, where and
, respectively. Here are determined the structurally stable
patterns of geometric mean curvature lines near the umbilic points, parabolic
curves and geometric mean curvature cycles, the periodic leaves of the
foliations. The genericity of these patterns is established. This provides the
three essential local ingredients to establish sufficient conditions, likely to
be also necessary, for Geometric Mean Curvature Structural Stability. This
study, outlined at the end of the paper, is a natural analog and complement for
the Arithmetic Mean Curvature and Asymptotic Structural Stability of immersed
surfaces studied previously by the authors.Comment: 21 pages, 5 figures. To appear in Annales de la Faculte de Sciences
de Toulous
Euler Integration of Gaussian Random Fields and Persistent Homology
In this paper we extend the notion of the Euler characteristic to persistent
homology and give the relationship between the Euler integral of a function and
the Euler characteristic of the function's persistent homology. We then proceed
to compute the expected Euler integral of a Gaussian random field using the
Gaussian kinematic formula and obtain a simple closed form expression. This
results in the first explicitly computable mean of a quantitative descriptor
for the persistent homology of a Gaussian random field.Comment: 21 pages, 1 figur
Minimal surfaces in S^3: a survey of recent results
In this survey, we discuss various aspects of the minimal surface equation in
the three-sphere S^3. After recalling the basic definitions, we describe a
family of immersed minimal tori with rotational symmetry. We then review the
known examples of embedded minimal surfaces in S^3. Besides the equator and the
Clifford torus, these include the Lawson and Kapouleas-Yang examples, as well
as a new family of examples found recently by Choe and Soret. We next discuss
uniqueness theorems for minimal surfaces in S^3, such as the work of Almgren on
the genus 0 case, and our recent solution of Lawson's conjecture for embedded
minimal surfaces of genus 1. More generally, we show that any minimal surface
of genus 1 which is Alexandrov immersed must be rotationally symmetric. We also
discuss Urbano's estimate for the Morse index of an embedded minimal surface
and give an outline of the recent proof of the Willmore conjecture by Marques
and Neves. Finally, we describe estimates for the first eigenvalue of the
Laplacian on a minimal surface.Comment: Published pape
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