Lectures Given At Summer School Of The Centro Internazionale Matematico Estivo

The Centro Internazionale Matematico Estivo (CIME) is a summer school that brings together mathematicians from all over the world to learn about the latest advances in mathematics. The school is held in Italy each year, and it offers a variety of courses on topics ranging from algebra to analysis to geometry. The goal of CIME is to provide a stimulating and intellectually challenging environment for mathematicians to learn and grow.

In 2019, CIME offered a course on "Lectures on Geometric Flows". The course was taught by Professor Robert J. McCann, and it covered a variety of topics related to geometric flows, including the Ricci flow, the mean curvature flow, and the Willmore flow. Professor McCann is a leading expert in geometric flows, and his lectures were very well-received by the students.

Non-linear Continuum Theories: Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Bressanone ... Schools, 36) (English and Italian Edition)

by Jane Yolen

4.3 out of 5

Language	:	English, Italian
File size	:	12651 KB
Screen Reader	:	Supported
Enhanced typesetting	:	Enabled
Print length	:	331 pages
Paperback	:	358 pages
Item Weight	:	1.13 pounds
Dimensions	:	6.1 x 0.82 x 9.25 inches

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The following article is a summary of the lectures given by Professor McCann at the CIME summer school in 2019.

The Ricci Flow

The Ricci flow is a geometric flow that evolves a Riemannian metric on a manifold according to the following equation:

$$\frac{\partial g}{\partial t}= -2 \text{Ric}(g)$$

where $g$ is the Riemannian metric, $t$ is the time parameter, and $\text{Ric}$ is the Ricci curvature. The Ricci flow was first introduced by Richard S. Hamilton in 1982, and it has since become a powerful tool for studying the geometry of Riemannian manifolds.

The Ricci flow has a number of interesting properties. First, it is a parabolic flow, which means that it is well-posed and has a unique solution for any initial metric. Second, the Ricci flow is a gradient flow, which means that it decreases a certain energy functional on the space of Riemannian metrics. Third, the Ricci flow is a preserving flow, which means that it preserves many important geometric properties of the manifold.

The Ricci flow has been used to solve a number of important problems in Riemannian geometry, including the Poincaré conjecture, the Thurston geometrization conjecture, and the Calabi-Yau manifold conjecture.

The Mean Curvature Flow

The mean curvature flow is a geometric flow that evolves a hypersurface in a Riemannian manifold according to the following equation:

$$\frac{\partial x}{\partial t}= H\nu$$

where $x$ is the hypersurface, $t$ is the time parameter, $H$ is the mean curvature of $x$, and $\nu$ is the outward unit normal to $x$. The mean curvature flow was first introduced by John W. Morgan in 1984, and it has since become a powerful tool for studying the geometry of hypersurfaces.

The mean curvature flow has a number of interesting properties. First, it is a parabolic flow, which means that it is well-posed and has a unique solution for any initial hypersurface. Second, the mean curvature flow is a gradient flow, which means that it decreases a certain energy functional on the space of hypersurfaces. Third, the mean curvature flow is a preserving flow, which means that it preserves many important geometric properties of the hypersurface.

The mean curvature flow has been used to solve a number of important problems in geometry, including the Dehn surgery theorem, the Willmore conjecture, and the Minkowski problem.

The Willmore Flow

The Willmore flow is a geometric flow that evolves a Riemannian metric on a closed surface according to the following equation:

$$\frac{\partial g}{\partial t}= -2 W\text{Ric}(g)$$

where $g$ is the Riemannian metric, $t$ is the time parameter, $W$ is the Willmore energy, and $\text{Ric}$ is the Ricci curvature. The Willmore flow was first introduced by T. J. Willmore in 1965, and it has since become a powerful tool for studying the geometry of Riemannian surfaces.

The Willmore flow has a number of interesting properties. First, it is a parabolic flow, which means that it is well-posed and has a unique solution for any initial metric. Second, the Willmore flow is a gradient flow, which means that it decreases a certain energy functional on the space of Riemannian metrics. Third, the Willmore flow is a preserving flow, which means that it preserves many important geometric properties of the surface.

The Willmore flow has been used to solve a number of important problems in geometry, including the Willmore conjecture, the Gauss-Bonnet theorem, and the uniformization theorem.

The Ricci flow, the mean curvature flow, and the Willmore flow are three of the most important geometric flows in mathematics. These flows have been used to solve a number of important problems in geometry

Non-linear Continuum Theories: Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Bressanone ... Schools, 36) (English and Italian Edition)

by Jane Yolen

4.3 out of 5

Language	:	English, Italian
File size	:	12651 KB
Screen Reader	:	Supported
Enhanced typesetting	:	Enabled
Print length	:	331 pages
Paperback	:	358 pages
Item Weight	:	1.13 pounds
Dimensions	:	6.1 x 0.82 x 9.25 inches

FREE