Convex bodies: the brunn–minkowski theory

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2F60590931937BF33789B5A986069C6D/S0025557200107697a.pdf/div-class-title-span-class-italic-curves-and-singularities-span-by-j-w-bruce-and-p-j-giblin-pp-222-1984-isbn-0-521-24945-7-hardback-25-27091-x-paperback-8-95-cambridge-university-press-div.pdf

Curves and singularities, by J. W. Bruce and P. J. Giblin. Pp 222. 1984. ISBN 0-521-24945-7 (Hardback) £25, 27091-X (Paperback) £8·95 (Cambridge University Press)

Singularities of caustics and wave fronts

Preface List of symbols 1. Morphology of a crystal surface 2. Surface free energy, step free energy, and chemical potential 3. The equilibrium crystal shape 4. Growth and dissolution crystal shapes: Frank's model 5. Crystal growth: the abc 6. Growth and evaporation of a stepped surface 7. Diffusion 8. Thermal smoothing of a surface 9. Silicon and other semiconducting materials 10. Growth instabilities of a planar front 11. Nucleation and the adatom diffusion length 12. Growth roughness at long lengthscales in the linear approximation 13. The Kardar-Parisi-Zhang equation 14. Growth without evaporation 15. Elastic interactions between defects on a crystal surface 16. General equations of an elastic solid 17. Technology, crystal growth and surface science Appendices References Index.

Physics of Crystal Growth

This is a survey article on the spherical method for studying Wulff shapes and related topics. The spherical method, which seems less common, is a powerful tool to study Wulff shapes and their related topics. It is verified how powerful the spherical method is by various results which seem difficult to be obtained without using the method. In this survey, the spherical method is explained in detail, and results obtained by using the spherical method until April 2016 are explained as well.

Spherical method for studying Wulff shapes and related topics

Abstract We consider finitely determined map germs f : (ℝ3, 0) → (ℝ2, 0) with f –1(0) = {0} and we look at the classification of this kind of germ with respect to topological equivalence. By Fukuda's cone structure theorem, the topological type of f can be determined by the topological type of its associated link, which is a stable map from S 2 to S 1. We define a generalized version of the Reeb graph for stable maps γ : S 2 → S 1, which turns out to be a complete topological invariant. If f has corank 1, then f can be seen as a stabilization of a function h 0: (ℝ2, 0) → (ℝ, 0), and we show that the Reeb graph is the sum of the partial trees of the positive and negative stabilizations of h 0. Finally, we apply this to give a complete topological description of all map germs with Boardman symbol Σ 2, 1.

/pdf/the-reeb-graph-of-a-map-germ-from-3-to-2-with-isolated-zeros-5d00hq31ig.pdf

The Reeb Graph of a Map Germ from ℝ3 to ℝ2 with Isolated Zeros

In this paper, it is shown that the set consisting of stable convex integrands $S^n\\to \\mathbb{R}_+$ is open and dense in the set consisting of $C^\\infty$ convex integrands with respect to Whitney $C^\\infty$ topology. Moreover, an application of the proof of this result is also shown.

/pdf/stability-of-c-convex-integrands-4z97chr9pc.pdf

Stability of c∞ convex integrands

Topological classification of circle-valued simple morse-bott functions

We consider the topological classification of finitely determined map germs $$[f]:(\mathbb {R}^3,0)\rightarrow (\mathbb {R}^2,0)$$
 with $$f^{-1}(0)\ne \{0\}$$
 . The case $$f^{-1}(0) = \{0\}$$
 was treated in another recent paper by the authors. The main tool used to describe the topological type is the link of [f], which is obtained by taking the intersection of its image with a small sphere $$S^1_\delta $$
 centered at the origin. The link is a stable map $$\gamma _f:N\rightarrow S^1$$
 , where N is diffeomorphic to a sphere $$S^2$$
 minus 2L disks. We define a complete topological invariant called the generalized Reeb graph. Finally, we apply our results to give a topological description of some map germs with Boardman symbol $$\Sigma ^{2,1}$$
 .

/pdf/the-reeb-graph-of-a-map-germ-from-mathbb-r-3-r-3-to-mathbb-r-3w3dziggjl.pdf

The Reeb Graph of a Map Germ from $$\mathbb {R}^3$$ R 3 to $$\mathbb {R}^2$$ R 2 with Non Isolated Zeros

In this paper, we investigate simultaneous properties of a convex integrand $\gamma$ and its dual $\delta$. The main results are the following three. 
(1) For a $C^\infty$ convex integrand $\gamma: S^n\to \mathbb{R}_+$, its dual convex integrand $\delta: S^n\to \mathbb{R}_+$ is of class $C^\infty$ if and only if $\gamma$ is a strictly convex integrand. 
(2) Let $\gamma: S^n\to \mathbb{R}_+$ be a $C^\infty$ strictly convex integrand. Then, $\gamma$ is stable if and only if its dual convex integrand $\delta: S^n\to \mathbb{R}_+$ is stable. 
(3) Let $\gamma: S^n\to \mathbb{R}_+$ be a $C^\infty$ strictly convex integrand. Suppose that $\gamma$ is stable. Then, for any $i$ $(0\le i\le n)$, a point $\theta_0\in S^n$ is a non-degenerate critical point of $\gamma$ with Morse index $i$ if and only if its antipodal point $-\theta_0\in S^n$ is a non-degenerate critical point of the dual convex integrand $\delta$ with Morse index $(n-i)$.

Erica Boizan Batista

Papers

The Reeb Graph of a Map Germ from ℝ3 to ℝ2 with Isolated Zeros

Stability of c∞ convex integrands

Topological classification of circle-valued simple morse-bott functions

The Reeb Graph of a Map Germ from $$\mathbb {R}^3$$ R 3 to $$\mathbb {R}^2$$ R 2 with Non Isolated Zeros

Simultaneous smoothness and simultaneous stability of a $C^\infty$ strictly convex integrand and its dual