Total curvature of submanifolds
Event details
| Date | 10.12.2014 |
| Hour | 16:30 › 17:30 |
| Speaker | Wolfgang Kühnel (Stuttgart) |
| Location |
GR A3 30
|
| Category | Conferences - Seminars |
Geometry and Dynamics Seminar
Abstract: The classical Gauss-Bonnet theorem states that the total (Gaussian) curvature of a compact surface equals its Euler characteristic. The Cohn-Vossen inequality on the total curvature gives an estimate for the Gauss-Bonnet defect that arises for complete, non-compact surfaces. The nature of this defect was further studied by Shiohama, Wintgen and others. For minimal surfaces it is known to be `quantized' by integer multiples of $2\pi$.
In higher dimensions this inequality is no longer valid. We present a simple 4-dimensional counterexample. Several sufficient intrinsic or extrinsic criteria are known that imply a vanishing defect. For odd-dimensional hypersurfaces the situation is completely different since the total curvature fails to be a topological invariant even in the compact case. Instead it depends on the regular homotopy class of the immersion. Here the simplest case is the Whitney-Graustein theorem on closed curves.
In the talk we study the difference term for even-dimensional complete submanifolds of Euclidean spaces of an appropriate type. It turns out that the Gauss-Bonnet defect is an intrinsic quantity of the `set at infinity' or the `ideal boundary'. This allows to discuss the condition for the total curvature to be stationary and a possible quantization of the total curvature in this case.
Abstract: The classical Gauss-Bonnet theorem states that the total (Gaussian) curvature of a compact surface equals its Euler characteristic. The Cohn-Vossen inequality on the total curvature gives an estimate for the Gauss-Bonnet defect that arises for complete, non-compact surfaces. The nature of this defect was further studied by Shiohama, Wintgen and others. For minimal surfaces it is known to be `quantized' by integer multiples of $2\pi$.
In higher dimensions this inequality is no longer valid. We present a simple 4-dimensional counterexample. Several sufficient intrinsic or extrinsic criteria are known that imply a vanishing defect. For odd-dimensional hypersurfaces the situation is completely different since the total curvature fails to be a topological invariant even in the compact case. Instead it depends on the regular homotopy class of the immersion. Here the simplest case is the Whitney-Graustein theorem on closed curves.
In the talk we study the difference term for even-dimensional complete submanifolds of Euclidean spaces of an appropriate type. It turns out that the Gauss-Bonnet defect is an intrinsic quantity of the `set at infinity' or the `ideal boundary'. This allows to discuss the condition for the total curvature to be stationary and a possible quantization of the total curvature in this case.
Practical information
- General public
- Free
Organizer
- Sonja Hohloch