Regular homotopy and total curvature I: circle immersions into surfaces

Imagine a nearly horizontal, statistically uniform, random surface ζ(x, y), Gaussian in the sense that the second derivatives , , have a normal joint distribution. The problem considered is the statistical distribution of the quantitywhere J and Ω denote the mean curvature and total curvature of the surface, respectively, and ν is a constant parameter.

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Curvature-dependent energies: a geometric and analytical approach

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210516000202 ◽

2017 ◽

Vol 147 (3) ◽

pp. 449-503 ◽

Cited By ~ 1

Author(s):

Emilio Acerbi ◽

Domenico Mucci

Keyword(s):

Bounded Variation ◽

Analytical Approach ◽

Total Curvature ◽

Representation Formula ◽

Energy Functional ◽

Continuous Functions ◽

Explicit Representation ◽

Functions Of Bounded Variation ◽

One Dimensional ◽

Relaxed Energy

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

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Immersions with non-zero normal vector fields

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100070961 ◽

1992 ◽

Vol 112 (2) ◽

pp. 281-285 ◽

Cited By ~ 1

Author(s):

Bang-He Li ◽

Gui-Song Li

Keyword(s):

Vector Fields ◽

Normal Vector ◽

Homotopy Classes ◽

Natural Action ◽

Singularity Method ◽

One To One ◽

Regular Homotopy

Let M be a smooth n-manifold, X be a smooth (2n − 1)-manifold, and g:M → X be a map. It was proved in [6] that g is always homotopic to an immersion. The set of homotopy classes of monomorphisms from TM into g*TX, which is denoted by Sg, may be enumerated either by the method of I. M. James and E. Thomas or by the singularity method of U. Koschorke (see [1] and references therein). When the natural action of π1(XM, g) on Sg is trivial, for example, if X is euclidean, the set Sg is in one-to-one correspondence with the set of regular homotopy classes of immersions homotopic to g (see e.g. [4]).

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