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

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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 $$0. Is it undoubted that $$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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