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 $0<r<1$. 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).

dg.differential geometry – Do smooth maps with nowhere-maximal rank have small image?

I’m trying to better understand the concept of “maps with small image” as used by Lipyanskiy in his construction of “geometric homology” in https://arxiv.org/abs/1409.1121. Lipyanskiy utilizes manifolds with corners, but for the purposes of this question I think it suffices to stick to ordinary manifolds, which we assume to be second countable.

By definition, a smooth map of manifolds $f: Wto M$ has small image if there is another smooth map of manifolds $g: Tto M$ such that $dim(T)<dim(W)$ and $f(W)subset g(T)$. I’m interested in alternative formulations of this condition. In particular, it seems reasonable to conjecture that this condition is equivalent to the map $f$ having less than full rank at all points.

In fact, I’m pretty sure that having small image implies that $f$ is nowhere of full rank: otherwise $f$ will be an immersion at some point and so the image will have dimension at least $dim(W)$. Then I believe the argument about Hausdorff dimension from this question about space filling curves implies that $f(W)$ cannot be covered by a smooth map with domain of smaller dimension: Proof that no differentiable space-filling curve exists

So my question really comes down to the converse: if $f$ nowhere achieves maximal rank, does it have small image?

I would also be interested in any other equivalent conditions to having small image.

Thanks!

dg.differential geometry – Does the continuous mapping space between topological manifolds always admit a Banach manifold structure?

Let $M$ and $N$ be smooth, i.e. $C^infty$, manifolds. Suppose that $M$ is compact. Then for every $k geq 0$ it is well known that $$C^k(M,N)$$ admits the structure of a smooth Banach manifold. I am interested in the continuous case, i.e. $k = 0$. I think that then one can drop the regularity of $M$ and $N$ and still get a Banach manifold $$C(M,N)$$ as one constructs the charts via an exponential map. My question: Is $C(M,N)$ a Banach manifold also in the case where $M$ and $N$ are merely topological manifolds, i.e. $C^0$ manifolds? I mean if $M$ and $N$ admit a $C^1$ structure or equivalently a $C^infty$ structure, then the statement is obvious, but this is not always the case.

dg.differential geometry – Hodge decomposition seems to say harmonic and exact is zero (and possibly (harmonic and co-exact) and (exact and co-exact))

(asked on maths se. no answers.)

Part of Hodge Decomposition Theorem says that for a compact oriented Riemannian (smooth) $m$−manifold $(M,g)$ (I think M need not be connected, but you may assume connected if need be or you want) and for a smooth $k$-form $omega$, i.e. $omega in Omega^k(M)$

A. $omega$ decomposes into exact, co-exact and harmonic: $omega =$ (respectively) $omega_d oplus omega_delta oplus omega_Delta $$in Omega^k(M) = B^kM bigoplus mathscr H^k(M,g) bigoplus image(delta_{k+1})$

B. $omega$ is zero if $omega$ is at least 2 of the ff: harmonic, exact, co-exact. Specifically:

  • B.1. ‘harmonic and exact is zero’

  • B.2. ‘harmonic and co-exact is zero’

  • B.3. ‘exact and co-exact is zero’

C. I think (B.1) is equivalent to the injectivity of the map $phi: mathscr H^k(M,g) := ker(Delta_k) to$$ H^k_{dR}M := frac{Z^kM}{B^kM}$, which maps a harmonic form to its de rham cohomology class, $phi(omega)=(omega):=omega + B^kM$ (or ‘$omegabmod B^kM$‘).

Proof: $phi$ is the restriction of $Phi:Z^kM to H^k_{dR}M$ (from closed $k$-forms) to harmonic $k$-forms. Then $ker(phi) = ker(Phi) cap Domain(phi) = B^kM cap mathscr H^k(M,g) = $$image(d_{k-1}) cap ker(Delta_k)$. QED

  • C.1. Note on notation: I personally would like to have $phi$ and $Phi$ to have subscripts $k$ since their domains and ranges depend on $k$, but I’m omitting the subscript $k$ because I’m following notation in the following powerpoint.

  1. What exactly is going on in slide 46? This is part of proving that $phi$ is injective.
  • 1.1. What I think: Here, Ryan Vaughn seems to conclude, for the decomposition of harmonic $omega = omega_d oplus omega_delta oplus omega_Delta$, that $omega_{Delta}$ is zero because $omega_{Delta}$ is exact and because of (B.1). However, I really don’t think decomposition is necessary because the other components of $omega$ will obviously be zero.

  • 1.2. I assume that the decomposition itself (I mean without the ‘orthonormal’ part) doesn’t rely on (B.1) (or (B.2)-(B.3)), otherwise I think this would be circular. I’m guessing that (B.1) (or (B.2)-(B.3)) is used in the ‘ortho(normal)’ part of the decomposition theorem. Thus, it makes sense to talk about $omega = omega_d oplus omega_delta oplus omega_Delta$ even if we don’t know (B.1) (or (B.2)-(B.3)).

  1. How does Ryan Vaughn prove (B.1), if Ryan Vaughn did? Otherwise, how does one prove (B.1) (of course without assuming the full Hodge Decomposition Theorem; soooo of course proving (B.1) is part of proving Hodge Decomposition Theorem)?

What I tried so far:

2.1 – I think the following is somehow relevant, if true:

a smooth $k$-form is harmonic if and only if closed and co-closed, i.e. $ker (Delta_k) = $$ ker(d_k) cap ker(delta_k) = Z^kM cap ker(delta_k)$

2.2 – If true: It seems ‘harmonic and exact’ is equivalent to ‘co-closed and exact’. Please explain how to show ‘co-closed and exact is zero’.

dg.differential geometry – Intersection of self-shrinkers

I have a problem regarding a statement in the paper Smooth compactness of self-shrinkers by Colding and Minicozzi.

In the article, they define a surface $Sigma$ in $mathbb R^3$ to be a self-shrinker if its mean curvature $H$ and the outer normal $n$ satisfy
$$H=frac{langle x,nrangle}2,$$
where $x$ is the position vector. In its corollary, it asserts, “Every self-shrinker must intersect the closed ball bounded by the spherical self-shrinker, which follows from the maximum principle since the associated MCF’s both disappear at the same point in space and time $(0,0).$

I think the spherical self-shrink is the sphere centered at the origin with radius $2,$ which is a self-shrinker by definition. However, I don’t understand how to use the maximum principle to derive that it intersects with any other self-shrinkers. Does this follows by something like avoidance principle for MCF? (But I think intersecting at the singularity doesn’t violate that principle…)

Any ideas or comments are appreciated!

dg.differential geometry – On the orbit of a Frechet Lie group action

Suppose that $G$ is a Fréchet Lie group acting on a Fréchet manifold $X$.
Fix $xin X$ and let $alpha(t)$ be a smooth path in $X$ such that
$$
begin{cases}
alpha(0)=x\
alpha(t)in Gcdot x
end{cases}.
$$

Also denote $rho_{x}:Grightarrow X:gmapsto gcdot x$. Is it true that $alpha'(0)in text{Im}(d_{e}rho_{x})$?

In the finite dimensional setting, this is clearly the case: the orbit $Gcdot x$ is a weakly embedded submanifold of $X$, hence $alpha(t)$ is also smooth as a curve in $Gcdot x$. Consequently $$alpha'(0)in T_{x}(Gcdot x)=text{Im}(d_{e}rho_{x}).$$

In the case that is of interest to me, $G=Diff(M)$ is the space of diffeomorphisms of a compact manifold $M$, and $X$ is the Fréchet space of rank $k$ – distributions $Gamma(Gr_{k}(M))$.

dg.differential geometry – On the perturbation of one vertex of an n-simplex so that point the given point on one of its face gets into its interior

I have a closed cube $Q_{0}=(0,1)^l$ and half-plane $H={(z_1,z_2,…,z_l) in mathbb{R}^l : z_1 + z_2 +…+ z_l > alpha }$ in Euclidean space $mathbb{R}^l$ with $0<alpha<1$. Consider the open convex set

$C={(z_1,z_2,…,z_l) in mathbb{R}^l : z_1 + z_2 +…+ z_l > alpha, 0<z_j < 1, j=1,2,…,l }$ in $mathbb{R}^l$.

Also, suppose I have $l+1$ points, $vec{z_1},vec{z_2},…,vec{z_l},vec{z_{l+1}}$ so that each point is in the intersection of a different face of $Q_0$ with $ H$ so that they do not lie on the same affine hyperplane (meaning these $l+1$ pants form $l$-simplex in $mathbb{R}^l$).

Now, consider point $vec{z} in C$ which also lie on the open line segment $(vec{z_1}, vec{z_2})$.

My question is to provide some argument which guarantees that I can perturb $vec{z_1}$ (or all $ vec{z_{i}}, i=1,2,…,l$) on the face containing it to, say, $vec{z_{1, epsilon}}$ so that the given point $vec{z}$ now belong to the open convex hull of $ { vec{z_{1, epsilon}},vec{z_2},…,vec{z_l},vec{z_{l+1}} } $.

Thanks so much, in advance.

dg.differential geometry – question about the book “Holomorphic Morse Inequalities” by Marinescu-Ma

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dg.differential geometry – What would be a good introductory reference for learning jet-bundle theory?

I am interested in learning the theory of Jet bundles, and am aware of the standard reference “The geometry of jet bundles” by D. J. Saunders. However this appears to be a detailed book, suitable for those who wish to specialise in this area. Can somebody recommend a relatively more introductory book (for a reader who knows the necessary differential geometric pre-requisites for learning this subject, but has never encountered Jet bundles) ? Thanks so much !

dg.differential geometry – Extensions of minimal hypersurfaces

Let $B subset mathbf{R}^{n+1}$ be the unit ball, and $M subset B$ be a minimal hypersurface. By this we mean that $M$ is an embedded $n$-dimensional submanifold with vanishing mean curvature. We allow for the closure $overline{M}$ of $M$ in $B$ to not be embedded, write $mathrm{sing} , M = overline{M} setminus M$ and call this the singular set of $M$. However this is assumed to be small enough for $M$ to be stationary in $B$: compactly supported deformations $X in C_c^1(B;mathbf{R}^{n+1})$ do not change the area of $M$ up to first order. For example one might take $n geq 2$ and consider a surface $M$ embedded outside the origin with $mathrm{sing} , M = { 0 }$.

Question 1. Are there conditions that allow the extension of $M$ to a globally defined immersed minimal hypersurface in $mathbf{R}^{n+1}$? That is, when is there a minimal hypersurface $tilde{M}$ in $mathbf{R}^{n+1}$ (immersed away from a small singular set) with $tilde{M} cap B = M$?

Let me make some remarks summarising my own conclusions.

  • The Cauchy–Kovalevskaya theorem could be relevant, but I am not sure whether this can be used to construct a globally defined extension. Moreover, one would have to worry about pieces coming together and meeting tangentially.
  • This is not a purely PDE-theoretic question. If one considers the case where $M$ is the graph of a smooth function $u$ defined on $mathbf{R}^n cap B$—this function satisfies the (quasi-linear) minimal surface equation—then it is not hard to see that $u$ can in general not be extended to a globally defined function $tilde{u}: mathbf{R}^n to mathbf{R}$. The Bernstein theorem is one way to see this, but also simple examples can be constructed using a suitable portion of the catenoid.
  • One can use the unique continuation property of minimal surfaces against the question, by taking $M$ to be a portion of a known surface. For example, by taking $M$ to be an embedded portion of an immersed minimal surface $tilde{M}$ one can see that one cannot hope for a globally defined and embedded minimal extension. Moreover, if one chooses $M$ to be portion of a singly-periodic Scherk surface one sees that $tilde{M}$ may have unbounded area growth: $lim_{R to infty} mathcal{H}^n(tilde{M} cap B_R)/R^n = infty$.

I am especially interested in the case where $M$ is one of the surfaces constructed by Caffarelli–Hardt–Simon. These are defined in $B$, embedded outside the origin, where they are prescribed to be tangent to a given minimal cone $mathbf{C}$.

Question 2. How does the answer change if $M$ is one of those surfaces? Is there $tilde{M}$ extending $M$, perhaps even with bounded area growth, that is with a constant $C > 0$ so that $mathcal{H}^n(tilde{M} cap B_R) leq C R^n$ for all radii $R > 1$?