Blog of Gabriel Peyré: https://twitter.com/gabrielpeyre
For example: Gradient flows: https://twitter.com/gabrielpeyre/status/1007865434320850944
The gradient field defines the steepest descent direction. The gradient flow dynamic defines a segmentation of the space into attraction bassins of the local minimizers. pic.twitter.com/wW0flPEWor
— Gabriel Peyré (@gabrielpeyre) 16/6/2018.
1. An example of the Gaussian curvature
Example 1 Compute the Gaussian curvature of sphere
The coefficients of the second fundamental form:
we compute coefficients of the first fundamental form:
These imply that
By the above computation, the curvature of sphere is
This paragraph we follow the book of Rudin . On linear functionals, there is special relationship between integration and linear functionals. In , a vector space, for any positive measure , the mapping
example, let be the set of all continuous functions on the unit interval . Then
has two properties:
so it is a linear functional on . Moreover, this is a positive linear functional: if then .
Consider a segment and the class of all with and or support of is . So we get . It is important that the length of related to the values of the functional .
There is an important theorem of F. Riesz, this illustrates to above event
Theorem 1 (F. Riesz) Let be a closed subset of . Then every positive linear functional on there corresponds a finite positive Borel measure on such that
 W. Rudin, Real and Complex analysis, Mc.Graw-Hill, 1970.
Let be a real analytic function with . Let and be a compact subset in . Then the (classical) \L ojasiewicz inequality asserts that:
- There exist such that
\noindent Let be a real analytic function with and . The \L ojasiewicz gradient inequality asserts that:
As a consequence, in (1), the order of zero of an analytic function is finite, and if is close to then is close to the zero set of . However, if is not compact, the latter is not always true and the inequality (1) does not always hold. The inequality (2) is similar to (1), it is not true in the case of is non-compact.
Two inequalities (1) and (2) have some special cases. For example, in the inequality (1), if has only isolated zero, i.e. , this implies . Hence, we have
On the other hand, being different from (2), we have another inequality:
There are some relations between and in complex case and real cases…