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

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

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

Let $M,N$ breathe two-dimensional flush, compact, linked, oriented Riemannian manifolds.

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

I’m exquisite 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 launch Euclidean domains, then $ Jf_n rightharpoonup Jf $ in $L^1(Okay)$ for any compact $Okay subset subset M$.

Without the idea $V(Jf_n le r) to 0$, this clearly would not maintain, plane when $f_n$ are conformal diffeomorphisms:

Take $M=N=mathbb{S}^2$. Let $s: mathbb{S}^2 to mathbb{R}^2 cup {infty}$ breathe the stereographic projection, and let $g_k(x) = okay x$ 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, flush 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 ceaseless duty. (asymptotically we squeeze greater and larger components of the sphere to a diminutive area across the pole).

So, we wouldn’t have feeble convergence of $Jf_n$ to $Jf=0$. (the

Jacobians converge as measures to a Dirac mass on the pole.) The query is that if by including the non-degeneracy constraint $V(Jf_n le r) to 0$ we regain this ‘Jacobian Rigidity’ underneath feeble convergence.

*(In my illustration of utility $r=frac{1}{4}$ however I do not cerebrate it issues).

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