On a family of $C^0$-convergent Riemann metrics

2012-05-15T20:04:06Z

I am dealing with the following concrete situation that could be familiar to Riemannian geometers more experienced than myself.

Suppose that $M$ is a smooth compact manifold of dimension $m$ and $g$ is a smooth Riemann metric on $M$. $\newcommand{\ve}{{\varepsilon}}$ $\newcommand{\pa}{\partial}$ Suppose that $(g^\ve)_{\ve>0}$ is a family of smooth Riemann metrics satisfying the following properties.

A. We know that for any $p\in M$ there exists an open neighborhood $U\ni p$ and local coordinates $x^1,\dotsc, x^m$ on $U$ such that

$$ g^\ve_{ij} \to g_{ij} $$

uniformly on the compacts of $U$, where

$$ g^\ve =\sum_{i,j} g^\ve_{ij}dx^idx^j,\;\;g=\sum_{i,j}g_{ij}dx^idx^j. $$

B. (Edited following Deane Yang's inquiry.) The note by $Gr_2(TM)$ the bundle of Grassmanians of $2$-planes in the tangent bundle. The sectional curvature $K^\ve$ can then be viewed as a function $K^\ve: Gr_2(TM)\to\mathbb{R}$. We know that there exists a smooth function $K^0: Gr_2(TM)\to \mathbb{R}$ such that $K^\ve\to K^0$ uniformly.

Question. Can we conclude that the function $K^0$ in B is the sectional curvature of $g$?

Answer by Anton Petrunin for On a family of $C^0$-convergent Riemann metrics

2012-05-16T01:58:32Z

Yes, it is true. Here is a sketch of proof, but I am sure that there is a simper way to see it.

Note that by Gauss–Bonnet, this is true in dimension 2.

Assume we know it in dimension $n-1$.

There is a construction of smooth convex hypersurfaces $H^\varepsilon$ in $(M,g^\varepsilon)$ which converge nicely to a smooth hypersurfaces $H$ in $(M,g^\varepsilon)$. "Nicely" means that one can apply the induction hypothesis for $H^\varepsilon\to H$. Taking many hypersurfaces like that plus linear algebra finishes the proof.

The construction. To construct $H^\varepsilon$ near $p\in M$, choose n points $a_1,a_2,\dots a_n$ such that in all the metrics the angles $\angle a_ipa_j\approx\pi/2$ for all $i\ne j$. Then define $$H^\varepsilon=\{\, x\in M\mid \sum\phi(|a_i-x|_{g^\varepsilon}-|a_i-p|_{g^\varepsilon})= 0 \,\}$$ where $\phi(0)=0$, $\phi'(0)=1$ and $\phi''\ll -2$.

On a family of $C^0$-convergent Riemann metrics - MathOverflow

On a family of $C^0$-convergent Riemann metrics

Answer by Anton Petrunin for On a family of $C^0$-convergent Riemann metrics