# 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… Expand

#### 27 Citations

On An Evolution Problem For Convex Curves

- 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… Expand

Conformal curvature flows: From phase transitions to active vision

- Mathematics
- ICCV 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… Expand

The Mathematics of F. J. Almgren Jr

- 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… Expand

Geometric heat equation and nonlinear diffusion of shapes and images

- Mathematics, Computer Science
- 1994 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. Expand

Rigidity Results for Hermitian-Einstein manifolds

- Physics, Mathematics
- 2013

A differential operator introduced by A. Gray on the unit sphere bundle of a K\"ahler-Einstein manifold is studied. A lower bound for the first eigenvalue of the Laplacian for the Sasaki metric on… Expand

Minimal surfaces based on the catenoid

- 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… Expand

Affine invariant scale-space

- Mathematics, Computer Science
- International 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. Expand

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, died on February 5, 1997 at the age of 63 as a result of… Expand

Dynamic active contours for visual tracking

- Mathematics, Computer Science
- IEEE 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. Expand

Active contours for visual tracking: a geometric gradient based approach

- Mathematics
- Proceedings of 1995 34th IEEE Conference on Decision and Control
- 1995

In this note, 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 metrics. This leads to a… Expand

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