Let $M$ be a compact subset of $\mathbb R^2$ with smooth boundary, and let $g$ be a Riemannian metric on $M$. If $g'$ is another Riemannian metric which is "close" to $g$, then they should have almost identical curvature profiles. I would like to prove a concrete estimate on the total difference of their curvatures in terms of the distance of $g'$ to $g$. Before I state the question precisely, I need to introduce some notation.


Write $\operatorname{Sym}$ for the space of symmetric $2\times 2$ real matrices, and let $\operatorname{SPD} \subseteq \operatorname{Sym}$ be those matrices which are also positive-definite. Consider the function space $\Omega = C^2(M, \operatorname{SPD})$. Denote partial derivatives of $g\_{ij} \in \Omega$ by additional subscripts following a comma, so that $\tfrac{\partial}{\partial x^k} g\_{ij} = g\_{ij,k}$, et cetera. Endow the space $\Omega$ with the norm $$\|g\| = \sup\_{x \in M} \max\_{i,j,k,l} \left\{|g\_{ij}(x)|, |g\_{ij,k}(x)|, |g\_{ij,kl}(x)| \right\},$$ so that it has the structure of an open cone within the Banach space $C^2(M, \operatorname{Sym})$.


Each $g \in \Omega$ defines a Riemannian structure on $M$ via the inner product $\langle v, g(x) v' \rangle$ for $v, v' \in T\_x M$. Let $K(g,x)$ be the scalar curvature of the metric $g$ at the point $x \in M$.


**What I want to prove:** For each $g \in \Omega$, there exist constants $C$ and $\epsilon$ so that if $g' \in \Omega$ with $\|g - g'\| < \epsilon$, then $$\sup\_{x \in M} \left| K(g,x) - K(g',x) \right| \le C\|g- g'\|.$$


My current approach to this is quite clunky, and involves calculating everything directly from the Christoffel symbols of the metrics. Is there a better, more geometric approach to this than brute force calculations?


I'm sure this type of lemma is well known to geometric analysts. Is a proof of a similar result written down somewhere?



How does curvature change under perturbations of a Riemannian metric?
This is a straightforward consequence of the fact that $K(x)$ is a continuous function of $g(x)$, $\partial g(x)$, and $\partial^2(g)$.


