# Some recent developments in differential geometry

@article{White1989SomeRD, title={Some recent developments in differential geometry}, author={Brian White}, journal={The Mathematical Intelligencer}, year={1989}, volume={11}, pages={41-47} }

Until recently differential geometry was the s tudy of fixed curves or surfaces in space and of abstract manifolds with fixed Riemannian metrics. Now geometers have begun to s tudy curves and surfaces that are subjected to various forces and that flow or evolve with time in response to those forces. Perhaps the simplest example (but already a very subtle one) is the curve-shortening flow. Consider a simple closed curve in the plane, and suppose that it moves so that the velocity at each point…

## 24 Citations

### On An Evolution Problem For Convex Curves

- Mathematics
- 2003

In this paper, we will investigate a new curvature flow for closed convex plane curves which shortens the length of the evolving curve but expands the area it bounds and makes the curve more and more…

### Conformal curvature flows: From phase transitions to active vision

- MathematicsICCV 1995
- 1995

In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new feature-based Riemannian…

### The Mathematics of F. J. Almgren Jr

- Mathematics
- 1998

Frederick Justin Almgren Jr., one of the world’s leading geometric analysts and a pioneer in the geometric calculus of variations, began his graduate work at Brown in 1958. It was a very exciting…

### Geometric heat equation and nonlinear diffusion of shapes and images

- Mathematics1994 Proceedings of IEEE Conference on Computer Vision and Pattern Recognition
- 1994

This work presents several properties of curvature deformation smoothing of shape: it preserves inclusion order, annihilates extrema and inflection points without creating new ones, decreases total curvature, satisfies the semi-group property allowing for local iterative computations, etc.

### Minimal surfaces based on the catenoid

- Mathematics
- 1990

DAVID HOFFMAN is Professor of Mathematics and Co-Director of the Geometry, Analysis, Numerics and Graphics Center (GANG) at the University of Massachusetts, Amherst. He earned his Ph.D. in…

### Affine invariant scale-space

- MathematicsInternational Journal of Computer Vision
- 2005

A newaffine invariant scale-space for planar curves is presented and the affine-invariant progressive smoothing property of the evolution equation is demonstrated as well.

### The mathematics of F. J. Almgren, Jr.

- Physics
- 1998

Frederick Justin Almgren, Jr, one of the world’s leading geometric analysts and a pioneer in the geometric calculus of variations, died on February 5, 1997 at the age of 63 as a result of…

### Dynamic active contours for visual tracking

- Computer ScienceIEEE Transactions on Automatic Control
- 2006

This work proposes an efficient, level set based approach for dynamic curve evolution, which addresses the artificial separation of segmentation and prediction while retaining all the desirable properties of the level set formulation.

### Active contours for visual tracking: a geometric gradient based approach

- Mathematics, Computer ScienceProceedings of 1995 34th IEEE Conference on Decision and Control
- 1995

This work analyzes geometric active contour models from a curve evolution point of view and proposes some modifications based on gradient flows relative to certain new metrics that lead to a novel snake paradigm in which the snake is attracted very naturally and efficiently to the desired feature.

### Boundary Value Problems for Minimal Surfaces

- Mathematics, Philosophy
- 1997

The aim of this survey is to describe some basic results on boundary value problems for minimal surfaces X : Ω → ℝ3 in three-dimensional Euclidean space. We are essentially concerned with questions…

## References

SHOWING 1-10 OF 22 REFERENCES

### Generalized rotational hypersurfaces of constant mean curvature in the Euclidean spaces. I

- Mathematics
- 1982

Among various basic local differential geometric invariants of a given hypersurface in the euclidean (n + l)-space, M n c E n+ι , the mean curvature, i.e. the trace of the second fundamental form, is…

### HYPERSURFACES MOVING WITH CURVATURE-DEPENDENT SPEED: HAMILTON-JACOBI EQUATIONS, CONSERVATION LAWS AND NUMERICAL ALGORITHMS

- Computer Science
- 1989

The goal of this paper is to show that algorithms based on direct parameterizations of the moving front face considerable difficulties, because such algorithms adhere to local properties of the solution, rather than the global structure.

### The structure of complete embedded surfaces with constant mean curvature

- Mathematics
- 1989

We consider complete, properly embedded surfaces Σ c R 3 which are of finite topological type and have constant nonzero mean curvature. Besides the round sphere the simplest such Σ are the…

### The motion of a surface by its mean curvature

- Physics
- 1978

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press to preserve the original texts of these important books while presenting them in durable paperback editions.

### Compact constant mean curvature surfaces in Euclidean three-space

- Mathematics
- 1987

The main subject of this paper is the construction of closed CMC surfaces of any genus g > 3 . The abbreviation "CMC surfaces" is used throughout the paper and stands for "properly immersed complete…

### Flow by mean curvature of convex surfaces into spheres

- Mathematics
- 1984

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],…

### The isoperimetric inequality

- Mathematics
- 1978

where A is the area enclosed by a curve C of length L, and where equality holds if and only if C is a circle. The purpose of this paper is to recount some of the most interesting of the many…

### Algorithms Based on Hamilton-Jacobi Formulations

- Computer Science
- 1988

New numerical algorithms, called PSC algorithms, are devised for following fronts propagating with curvature-dependent speed, which approximate Hamilton-Jacobi equations with parabolic right-hand-sides by using techniques from the hyperbolic conservation laws.

### Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. Final report

- Computer Science
- 1987

### The surfaces of Delaunay

- Mathematics
- 1987

In 1841 the astronomer/mathematician C. Delaunay isolated a certain class of surfaces in Euclidean space, representations of which he described explicitly [1]. In an appendix to that paper, M. Sturm…