# dg.differential geometry – Does higher integrability of Jacobians hold between manifolds when the Jacobians are concentrated?

$$newcommand{M}{mathcal{M}}$$
$$newcommand{N}{mathcal{N}}$$

Let $$M,N$$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds.

Let $$f_n rightharpoonup f$$ in $$W^{1,2}(M,N)$$ with $$Jf_n > 0$$ a.e., and suppose that the volume $$V({x in M , | , Jf_n le r}) to 0$$ when $$n to infty$$, for some $$0. Is it true that $$Jf_n rightharpoonup Jf$$ in $$L^1(M)$$?

I am fine with assuming that $$f_n$$ are Lipschits and injective and that $$V(f_n(M)) to V(N)$$.

The “higher integrability property of determinants” implies that if $$M,N$$ are open Euclidean domains, then $$Jf_n rightharpoonup Jf$$ in $$L^1(K)$$ for any compact $$K subset subset M$$.

Without the assumption $$V(Jf_n le r) to 0$$, this clearly doesn’t hold, even when $$f_n$$ are conformal diffeomorphisms:

Take $$M=N=mathbb{S}^2$$. Let $$s: mathbb{S}^2 to mathbb{R}^2 cup {infty}$$ be the stereographic projection, and let $$g_k(x) = k x$$ for $$x in R^2$$ (and $$g_n(infty) = infty$$.).

Set $$f_n = s^{-1} circ g_n circ s$$. $$f_k$$ are conformal, orientation preserving, smooth diffeomorphisms
and thus $$int_{mathbb{S}^2 }Jf_n=V(mathbb{S}^2 )$$. By conformality $$int_{mathbb{S}^2 } |Df_n|^2 =2int_{mathbb{S}^2 }Jf_n$$ is uniformly bounded, so $$f_n$$ is bounded in $$W^{1,2}$$, and converges to a constant function. (asymptotically we squeeze bigger and bigger parts of the sphere to a small region around the pole).

So, we do not have weak convergence of $$Jf_n$$ to $$Jf=0$$. (the
Jacobians converge as measures to a Dirac mass at the pole.) The question is if by adding the non-degeneracy constraint $$V(Jf_n le r) to 0$$ we recover this ‘Jacobian Rigidity’ under weak convergence.

*(In my case of application $$r=frac{1}{4}$$ but I don’t think it matters).