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


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 $0<r<1$. 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).