https://math.berkeley.edu/~sethian/Papers/sethian.osher.88.pdf

Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations

Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition)

We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.

/pdf/the-entropy-formula-for-the-ricci-flow-and-its-geometric-yzmm7lei64.pdf

The entropy formula for the Ricci flow and its geometric applications

Preface Through many centuries physics has been one of the most fruitful sources of inspiration for mathematics. As a consequence, mathematics has become an economic language providing a few basic principles which allow to explain a large variety of physical phenomena. Many of them are described in terms of partial diierential equations (PDEs). In recent years, however, mathematics also has been stimulated by other novel elds such as image processing. Goals like image segmentation, multiscale image representation, or image restoration cause a lot of challenging mathematical questions. Nevertheless, these problems frequently have been tackled with a pool of heuristical recipes. Since the treatment of digital images requires very much computing power, these methods had to be fairly simple. With the tremendous advances in computer technology in the last decade, it has become possible to apply more sophisticated techniques such as PDE-based methods which have been inspired by physical processes. Among these techniques, parabolic PDEs have found a lot of attention for smoothing and restoration purposes, see e.g. 113]. To restore images these equations frequently arise from gradient descent methods applied to variational problems. Image smoothing by parabolic PDEs is closely related to the scale-space concept where one embeds the original image into a family of subsequently simpler , more global representations of it. This idea plays a fundamental role for extracting semantically important information. The pioneering work of Alvarez, Guichard, Lions and Morel 11] has demonstrated that all scale-spaces fulllling a few fairly natural axioms are governed by parabolic PDEs with the original image as initial condition. Within this framework, two classes can be justiied in a rigorous way as scale-spaces: the linear diiusion equation with constant dif-fusivity and nonlinear so-called morphological PDEs. All these methods satisfy a monotony axiom as smoothing requirement which states that, if one image is brighter than another, then this order is preserved during the entire scale-space evolution. An interesting class of parabolic equations which pursue both scale-space and restoration intentions is given by nonlinear diiusion lters. Methods of this type have been proposed for the rst time by Perona and Malik in 1987 190]. In v vi PREFACE order to smooth the image and to simultaneously enhance semantically important features such as edges, they apply a diiusion process whose diiusivity is steered by local image properties. These lters are diicult to analyse mathematically , as they may act locally like a backward diiusion process. …

/pdf/anisotropic-diffusion-in-image-processing-z13u09qvsm.pdf

Anisotropic diffusion in image processing

The motion of surfaces by their mean curvature has been studied by Brakke [1] from the viewpoint of geometric measure theory. Other authors investigated the corresponding nonparametric problem [2], [5], [9]. A reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal. In this paper we take a more classical point of view: Consider a compact, uniformly convex w-dimensional surface M = Mo without boundary, which is smoothly imbedded in R. Let Mo be represented locally by a diffeomorphism

/pdf/flow-by-mean-curvature-of-convex-surfaces-into-spheres-2znctp13al.pdf

Flow by mean curvature of convex surfaces into spheres

is satisfied. Here H(p,ή is the mean curvature vector of the hypersurface Mt at F(/?, t). We saw in [7] that (1) is a quasilinear parabolic system with a smooth solution at least on some short time interval. Moreover, it was shown that for convex initial data Mo the surfaces Mt contract smoothly to a single point in finite time and become spherical at the end of the contraction. Here we want to study the singularities of (1) which can occur for nonconvex initial data. Our aim is to characterize the asymptotic behavior of Mt near a singularity using rescaling techniques. These methods have been used in the theory of minimal surfaces and more recently in the study of semilinear heat equations [3], [4]. An important tool of this approach is a monotonicity formula, which we establish in §3. Assuming then a natural upper bound for the growth rate of the curvature we show that after appropriate rescaling near the singularity the surfaces Mt approach a selfsimilar solution of (1). In §4 we consider surfaces Mt9 n > 2, of positive mean curvature and show that in this case the only compact selfsimilar solutions of (1) are spheres. Finally, in §5 we study the model-problem of a rotationally symmetric shrinking neck. We prove that the natural growth rate estimate is valid in this case and that the rescaled solution asymptotically approaches a cylinder.

/pdf/asymptotic-behavior-for-singularities-of-the-mean-curvature-2aqz0npq2w.pdf

Asymptotic-behavior for singularities of the mean-curvature flow

Let M be an asymptotically flat 3-manifold of nonnegative scalar curvature. The Riemannian Penrose Inequality states that the area of an outermost minimal surface N in M is bounded by the ADM mass m according to the formula |N |≤ 16πm 2 . We develop a theory of weak solutions of the inverse mean curvature flow, and employ it to prove this inequality for each connected component of N using Geroch’s monotonicity formula for the ADM mass. Our method also proves positivity of Bartnik’s gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry.

/pdf/the-inverse-mean-curvature-flow-and-the-riemannian-penrose-41awfzfdk1.pdf

The inverse mean curvature flow and the Riemannian Penrose Inequality

/pdf/interior-estimates-for-hypersurfaces-moving-by-mean-bs9qdu0dmc.pdf

Interior estimates for hypersurfaces moving by mean curvature

/pdf/mean-curvature-evolution-of-entire-graphs-gbkezhpjgk.pdf

Gerhard Huisken

Papers

Flow by mean curvature of convex surfaces into spheres

Asymptotic-behavior for singularities of the mean-curvature flow

The inverse mean curvature flow and the Riemannian Penrose Inequality

Interior estimates for hypersurfaces moving by mean curvature

Mean curvature evolution of entire graphs