measure theory – Question about a proposition in Munkres’s Analysis on Manifolds

I am reading through Munkres’s Analysis on Manifolds, and I get stuck in a proof of the lemma 18.1, that is stated as following:

Lema 18.1 Let $A$ be open in $mathbb{R}^n$; let $g:Ato mathbb{R}^n$ be a function of class $C^1$. If the subset $E$ of $A$ has measure zero in $mathbb{R}^n$, then $g(E)$ has measure zero in $mathbb{R}^n$.

He made out its proof in three steps. The first and second step are mentioned in the third, where he actually prove the theorem. Let me add some pictures of the third step.

enter image description here

enter image description here

(If you need pictures of the other two steps in order to solve the question above, let me know, please)

Note: A $delta$-neighborhood of a set $X$ is the union of all open cubes (in this case) with width $delta>0$ and centered at $xin X.$ The theorem 4.6 in that book states that every compact set $K$ that is contained in an open set $Usubset mathbb{R}^n$ has a $delta$-neighborhood contained in $U$.

So, the problem is here: When he covers the set $E_k$ by countably many cubes $D_i$ with certain properties, he asserts: Because $D_i$ has width less than $delta$, it is contained in $C_{k+1}$.

Why this is true? I mean, if each cube $D_i$ is centered at some point lying at $C_k$ it is clearly true, but we don’t know if this happens. I tried to give a proof that we can assume that each $D_i$ can be choosen in a way that is centered in $C_k$ but I couldn’t prove that.

Can you help me to justify that assertion on the book? Thanks in advance.

Improving regularity of the boundary of a convex set in Riemannian manifolds

Let $X$ be a geodesically complete Riemannian manifold (we may assume that $X$ is simply connected and negatively curved, although I don’t think it matters). Given a closed, convex subset $K subset X$, there is a result by Rolf Walter that the $epsilon$-neighbourhood of $K$ (denoted $K_{epsilon}$) has $C^{1,1}$-regular boundary, i.e. the topological boundary of $K_{epsilon}$ is a $C^{1,1}$-submanifold of $X$. I was wondering if the regularity can be improved if we allow for more deformations of the convex set. Specifically, I would like to know the following:

${bf Question:}$ Given a closed, convex subset $K subset X$ and $epsilon > 0$, does there exist a convex closed $K’$ such that $K subset K’ subset K_{epsilon}$, and $partial K’$ is a $C^2$-submanifold of $X$?}

Considering that Walter’s result goes back to the 70s (Rolf Walter, “Some analytical properties of geodesically convex sets”), this seems like a question whose answer should be known, but I’m struggling to find anything. All methods I could come up with to deform $K$ so that its boundary becomes more smooth while keeping the set convex rely on the second fundamental form, which requires that we make the boundary $C^2$ first.

I should empathize that I do not want to assume that $K$ is compact. (I’m fine assuming that there is a cocompact action by isometries on $K$ though.)

gt.geometric topology – Manifolds with trivial mapping class group and large $H^1$?

Are there smooth closed manifolds $M^n$ in every dimension $n geq 3$ with trivial mapping class groups and with $H^1(M^n;mathbb{Z}/2mathbb{Z})$ arbitrarily large?

I am under the impression that “generically” a manifold will have no mapping class group (but maybe I am totally mistaken). There are presumably lots of constructions of such manifolds with trivial mapping class groups. So I guess I’m hoping to hear of some such construction where $H^1(M^n;mathbb{Z}/2mathbb{Z})$ can get very large.

ag.algebraic geometry – Quotients of complex manifolds by symmetric group

Let $X$ and $Y$ be two complex manifolds of dimension $n$, $ngeq 2$. Denote by $Aut(X)$ and $Aut(Y)$ the group of bi-holomorphisms of $X$ and $Y$, respectively. Suppose the symmetric group on
$n$-symbols $S_n$ is contained in both $Aut(X)$ and $Aut(Y)$ such that

(1) $X/S_n$ and $Y/S_n$ are complex manifolds of dimension $n$;

(2) $X/S_n$ and $Y/S_n$ are bi-holomorphic.

Is it true that $X$ and $Y$ are bi-holomorphic?

ag.algebraic geometry – Manifolds with a Kähler deformation

Let $X$ be a compact complex non-Kähler manifold, then what conditions do we need to make it has a Kähler deformation? that is to say it can be deformed to a Kähler manifold.

Obviously not all the compact complex manifolds can be deformed to Kähler ones, for example, the Hopf surface, but certainly, there exist some non-Kähler manifolds which can be deformed to Kähler ones. For example, Hironaka has provided an example that except the central fiber, all the other fibers are projective manifolds, and the central fiber is a non-Kähler Moishezon manifold, so, conversely, for this Moishezon manifold, we can say it has a Kähler (even projective) deformation.

Then, are there any other examples of non-Kähler manifolds which has a Kähler deformation? or even a projective deformation? For example does a $partialbarpartial$-manifolds with trivial canonical bundle has a Kähler deformation? Has anyone think about it before? And what’s the latest progress on it?

ag.algebraic geometry – All compact complex manifolds with deformations unobstructed

I want to find out all the compact complex manifolds with deformations unobstructed, that is to say, for a compact complex manifold $X$, the local universal deformation space is smooth and isomorphic to an open set in $H^1(X,T_X)$.

There are 3 main classes which are already well-known:

1.Compact complex manifold $X$ with $H^2(X,T_X)=0$.
2.Calabi-Yau manifold(Compact Kähler manifold with canonical bundle trival)
3.$partialbarpartial$-complex symplectic manifold($partialbarpartial$-manifold with a nondegenerate $d$-closed $(2,0)$ form)

For the second class, it is proved by Tian and Todorov, and the Kähler condition can be weakened to the condition that the $partialbarpartial$-lemma holds(a manifold with $partialbarpartial$-lemma holds is called $partialbarpartial$-manifold), and the $partialbarpartial$ condition can be weakened to Frolicher spectral sequence degenerates at $E_1$ or the homotopy abelianness of the differential graded algebra $(Λ^{0,∗}(T_X), barpartial) $.(which one is weaker?)

For the third class, I’m not sure my description is equivalent to the one in KV19, since it’s the only source I know which talks about the unobstructed deformations of holomorphically symplectic manifolds. Actually in (KV19), the authors consider the holomorphically symplectic manifolds with Dolbeault cohomology group
$H^{0,2}_{bar ∂} (X) = H^2(X,mathcal O_X)$ generated by $∂$-closed $(0, 2)$-forms, and the condition that all $∂$-exact holomorphic 3-forms on $X$ vanish can be deduce by $partialbarpartial$-lemma.

My question is: apart from these 3 classes and their slightly generalizations, are there any other manifolds with deformations unobstructed?
(By the way, apart from the first class, the $partialbarpartial$ condition seems necessary for unobstructed deformations, I wonder whether
there exist manifolds with $H^2(X,T_X)=0$ but the $partialbarpartial$-lemma not hold? )

ag.algebraic geometry – Manifolds with $[,]:H^1(X,T_X)times H^1(X,T_X)rightarrow 0$

Let $X$ be a compact complex manifold, for arbitrary $phi_1,phi_2in H^1(X,T_X)$, if the Lie bracket $(,):H^1(X,T_X)times H^1(X,T_X)rightarrow H^2(X,T_X)$ always maps $phi_1,phi_2$ to zero, i.e.$(phi_1,phi_2)=0in H^2(X,T_X)$, then $X$ admits an unobstructed deformation?

As we know, if we assume $H^2(X,T_X)=0$, then of course the Lie bracket gives a map $(,):H^1(X,T_X)times H^1(X,T_X)rightarrow 0$, and for a Calabi-Yau manifold, by Tian-Todorov lemma and $partial barpartial$-lemma, the Lie bracket also maps $H^1(X,T_X)times H^1(X,T_X)$ to $0$, so, generally, if we assume that the Lie bracket always maps $H^1(X,T_X)times H^1(X,T_X)$ to $0$, then can we say $X$ must have an unobstructed deformation?

3 manifolds – Thurston Universe gates in Knots: which invariant is it?

Today I discovered this nice video of a lecture by Thurston:

https://youtu.be/daplYX6Oshc

in which he explains how a knot can be turned into a “fabric for universes”. For example, the unknot can be thought as a portal to Narnia, and when you pass again you switch back to Narnia. This forms in a sense a $mathbb{Z}/2mathbb{Z}$. Then he proceeds to explore what fabric one gets with the treefoil and you get an $S_3$. I am sure there is some real geometry behind but I can’t grasp how to translate a portal into something homotopic.

A way to formalize this would be the following. Take a knot $K$ in $mathbb{R}^3$. Fix a tubular neighborhood $N$ of $K$. For each point $x$ in the knot take the loop $L_x$ obtained as the sphere bundle of $N$ at $x$ (a small circle around $x$ that jumps into the portal). Then there exist a connected 3-manifold $M$ with a (finite?) cover $M to mathbb{R}^3 setminus K$ such that the “monodromy in small circles” around $L_x$ has order two for all $x$. Then we set the “group of universes” as the group of cover automorphisms.

I think this captures the previous idea in the following sense: to each locus of $mathbb{R}^3 setminus K$ we have $n$ counterimages that represent the different worlds. Some branch should be chosen to make distinguishing between worlds possible. The constraint on monodromy ensures that if you jump twice through the same portal (at least for the portals very close to the boundary) you get back.

Does such a manifold exist for all knots? Is this construction just some simplification of the fundamental group of the complement?

ag.algebraic geometry – Examples of complex manifolds with trivial Néron–Severi group?

$DeclareMathOperatorNS{NS}DeclareMathOperatorPic{Pic}$Let $X$ be a compact complex manifold, assume projective if you’d like. Define the Néron–Severi group to be the quotient $$NS(X) = Pic(X) / Pic^0(X).$$ Suppose that $Pic(X) = Pic^0(X) neq 0$. So all divisors are algebraically equivalent, and (by definition) the Picard number is zero.

Can we infer any geometric information from this constraint (does this constrain other invariants such as Kodaira dimension, curvature, etc.)? Are there examples of such $X$? Are there plenty of such examples?

smooth manifolds – Finitely connected orientable surface

Let $(M,g)$ be a finitely connected orientable complete Riemannian surface, that is, $M$ is homeomorphic to a compact orientable surface $Sigma$ minus $k geq 1$ points. Do you have references or a proof for the fact that $(M,g)$ is conformal to a compact orientable Riemann surface with $k geq 1$ disks deleted?

This is part of an argument in the paper “ On complete minimal surfaces with finite Morse index in three manifolds”, by Fischer-Colbrie.

argument