**7**

votes

**1**answer

697 views

### semi flat connections

Let $L\to V$ be complex line bundle and $F_{t}:V\to V$, $t\in [0,1]$, be a loop of diffeomorphisms, $F_0=F_1=$ identity.
For every $x\in V$, we get a loop $\gamma_x(t)=\{F_t(x)\}$ whose class in ...

**5**

votes

**1**answer

288 views

### Techniques to solve a non linear differential equation related to curvature

Many years ago, I considered the following non linear differential equation:
$y=y''.(1+y'^{2})^{-3/2}$
This equation expresses the equality between the value of a given function $y\in C^{2}(R)$ and ...

**8**

votes

**2**answers

149 views

### Interior periodic points of area preserving homeomorphisms of a pair of pants

A celebrated result of Franks shows that any area preserving homeomorphism of the closed annulus $A$ with at least one periodic point (possibly along the boundary) has infinitely many interior ...

**14**

votes

**1**answer

336 views

### List of Applications of the $\partial\overline{\partial}$-lemma

Quoting from Huybrecht's book Complex Geometry on the $\partial\overline{\partial}$-lemma for Kaehler manifolds:
Although it looks like a rather innocent technical statement, it is
crucial for ...

**2**

votes

**0**answers

101 views

### Semistability of a sheaf on nodal curve

Suppose $X$ is a projective, connected, nodal curve (can be reducible) over an algebraically closed field $k$ of arbitrary characteristic. Let $F$ be a pure sheaf on $X$ and denote by $\pi^{*}(F)$ its ...

**19**

votes

**1**answer

673 views

### Time averages and differentiability

Let $\varphi_t : M \rightarrow M$ be a smooth flow on a smooth manifold $M$. We may assume (although I'm not sure if this is important) that the flow preserves a smooth volume form on $M$. Given a ...

**0**

votes

**0**answers

51 views

### Splitting of totally geodesic Riemannian foliations

Let $\mathcal F$ be a non-singular Riemannian foliation on $(M,g)$ whose leaves are totally geodesic. Suppose further that the leaves are Riemannian products of irreducible manifolds $L=L_0\times ...

**0**

votes

**0**answers

78 views

### Mathematical consulting or bioinformatics related careers for mathematicians with good statistics and coding experience in West Europe [on hold]

Before I start, apologies if the question is very specific, but these are exactly what I want to be. I should mention that I already studied:
"Industry"/Government jobs for mathematicians
...

**3**

votes

**1**answer

559 views

### Does exist a $\varepsilon$-tubular neighborhood of a smooth complex quasi-affine algebraic variety

By a complex quasi-affine variety i mean the complement of an affine algebraic variety with respect to another algebraic variety, more precisely a quasi-affine algebraic variety is
$$V= ...

**0**

votes

**0**answers

76 views

### Mean curvature and submanifold

Consider $S^{N-1}$ the unit sphere and let us focus our attention on the cap
$$
G=S^{N-1}\cap\{x_N>0\}
$$
with boundary $\partial G= S^{N-2}\times\{0\}$: it is quite obvious to see that $G$ is a ...

**0**

votes

**1**answer

470 views

### Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?

Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me:
Is there any open Ricci-flat ALE 4-manifold other than ...

**2**

votes

**1**answer

135 views

### projectivity with assumption of big and semi-amplness

Let $X$ be a compact Kaehler manifold with $D$ be an effective divisor on $X$ such that $K_X+D$ is semi-ample and big then $X$ is projective?

**10**

votes

**2**answers

532 views

### Generalized smooth spaces and infinite dimensional manifolds

There is a theorem due to Losik which shows that the category of Frechet manifolds embeds fully-faithfully into diffeological spaces. (Diffeological spaces are concrete sheaves on the site of ...

**6**

votes

**1**answer

326 views

### The Hypercomplex Structure of $SU(3)$

(A) In this really stylish answer it is shown that one can define a family of complex structures $J_{\lambda}$ on the Lie group SU(3), dependent on the parameter $\lambda \in {\mathbb C}\backslash ...

**5**

votes

**0**answers

53 views

### Does the Hodge *-operator act on the tangent space at 0 to the space of integral (n-1)-cycles in a conformal manifold of dimension d=2n?

Suppose $M$ is a compact, oriented conformal manifold of even dimension $d=2n$.
Write ${\cal D}^{\mathit{int}}_{k}(M)$ for the space of integral
$k$-currents in $M$
and write ${\cal ...

**-1**

votes

**0**answers

70 views

### Embedded Minimal Surfaces in 3D Hyperbolic Space [closed]

My question concerns the embedding of minimal surfaces in the 3D hyperbolic space. The minimal surface is defined in terms of Weierstrass-Enneper representation.
Let us take now a slice (section) of ...

**13**

votes

**1**answer

197 views

### Does a Kähler manifold always admit a complete Kähler metric?

Every smooth manifold admits a complete Riemannian metric. In fact, every Riemannian metric is conformal to a complete Riemannian metric, see this note. What about in the Kähler case?
Does a ...

**4**

votes

**2**answers

525 views

### Alternative Almost Complex Structures

Originally posted on Maths Stack Exchange.
Let $V$ be a real vector space. An almost complex structure on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}_V$. An almost complex structure ...

**7**

votes

**4**answers

1k views

### Smoothing of the distance function on a Riemannian manifold

Suppose $(M,g)$ is a complete Riemannian manifold. $p\in M$ is a fixed point. $d_{p}(X)$ is the distance function defined by $p$ on M (i.e., $d_p(x)$=the distance between $p$ and $x$). Let ...

**4**

votes

**0**answers

61 views

### Geometrically-explicit upper bound for on-diagonal heat kernel

Let $M$ be a compact Riemannian manifold, and $K(t;z,w)$ the heat kernel associated to the usual Laplace-Beltrami operator on functions. There are results of the form
$$K(t;z,z) \leq ...

**15**

votes

**1**answer

1k views

### What is a Futaki invariant, what is the intuition behind it, and why is it important?

As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?

**0**

votes

**0**answers

109 views

### Are induced Riemannian metrics weakly continuous?

Let $\Omega \subseteq \mathbb{R}^d$ a nice bounded domain.
Let $F_n:\Omega \to \mathbb{R}^D$ be a sequence of smooth embeddings* in $W^{1,p}(\Omega,\mathbb{R}^D)$. Assume $F_n \rightharpoonup F$ in ...

**11**

votes

**2**answers

433 views

### Unexpected regularity of the distance from a $C^2$ submanifold

Let $\Gamma$ be a $C^2$ compact submanifold of $\mathbb{R}^n$. Consider the distance function $\delta$ from $\Gamma$. It is well known that, for sufficiently small $\varepsilon>0$, $\delta$ is ...

**1**

vote

**0**answers

98 views

### Question on Harmonic maps between Riemannian manifolds

In Theory of harmonic maps, main goal is to find minimum of Dirichlet energy function which is defined as follows:
$$E(f):=\int_M\|df\|dvol_g\qquad f:(M,g)\to(N,h).$$
In many Books such as Calculus ...

**7**

votes

**1**answer

424 views

### Which sections of $T^*M\odot T^*M$ have reproducing kernel “primitives”?

Given a smooth reproducing kernel $\kappa:M\times M\rightarrow \mathbb{R}$ on a manifold $M$, we can construct a section, $\alpha_{\kappa}$, of the symmetric tensor product $T^*M\odot T^*M$ by taking ...

**3**

votes

**1**answer

232 views

### How to tell if it's a Moishezon morphism

Suppose that $f \colon X\rightarrow S$ is a proper morphism of reduced and irreducible complex spaces and $f$ is a smooth deformation in the sense of Kodaira and Spencer. If we know each fiber $X_s$, ...

**7**

votes

**1**answer

102 views

### Properties of connection Laplacian on vector fields

Let $(M,g)$ be a simply-connected compact surface with boundary $\partial M$ and metric $g$. Let $N$ denote the outward unit normal on $\partial M$, $\nabla$ the Levi-Civita connection and $\Delta_g$ ...

**2**

votes

**0**answers

48 views

### On two functions with isodirectional gradients

Let $U\subset \mathbb{R}^n$ be open and $f,g:U \to \mathbb{R}$ be two $C^1$ functions whose gradients are always in the same direction, i.e. $\forall i,j \in \left\{1,...,n\right\}$
\begin{equation}
...

**24**

votes

**3**answers

1k views

### What's the supersymmetric analogue of the Monster group?

Bosonic string theory lives in 26 dimensions, and it gives a conformal field theory where the field is a map from a Riemann surface to $\mathbb{R}^{24}$. The Leech lattice $L$ is an even unimodular ...

**7**

votes

**0**answers

817 views

### The integral of torsion

I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...

**0**

votes

**0**answers

73 views

### Two questions about Li-Yau-Hamilton estimate

This question is from my question on mathematics.
Picture below is from 231 page . For to prove $Q\ge 0$ on $M \times (0,T)$,
$(\partial_t -\Delta)Q \ge 0$ and $Q\ge 0$ are needed to prove.But I ...

**0**

votes

**1**answer

95 views

### 3-form torsion and Cartan structural equations

First, my level of math isn't very high as I come from the physics world.
I am trying to understand the derivation of Cartan's 3-form torsion.
I've read Robert Bryant's answer in this thread:
Relating ...

**1**

vote

**0**answers

45 views

### Covering rough boundaries of closed sets in manifolds by charts

This question is a little vague, I'm afraid, because I'm not sure I expect there to be a complete answer; but there should be some sort of situations where it is possible.
Consider a Riemannian ...

**4**

votes

**1**answer

524 views

### Aysmptotic comparison of $L^2$ sections versus generating sections

Let $s_1,\ldots, s_k$ be linearly independent global holomorphic sections of a holomorphic line bundle $E$ over a compact algebraic manifold $X$, with volume form $\Omega$.
For $m$ large, let ...

**2**

votes

**0**answers

101 views

### Can infinitely many orbifolds be “added up” to form a fractal space?

Disclaimer: this question is rather vague and thus might not be suitable for this site. If so, feel free to tell me and I'll delete it.
Intuitively, an orbifold, from what I understand, is a ...

**4**

votes

**0**answers

55 views

### Uniform approximation of a continuous flow by a $\mathcal{C}^1$ flow

Setup: Consider a (smooth) compact Riemannian manifold $M$, whose distance is denoted by $d$. Let $\Phi$ be a continuous flow, namely a continuous application from $\mathbb{R} \times M $ to $M$ ...

**0**

votes

**1**answer

396 views

### Positivity of forms

I'm getting stuck with a proposition. Please can someone help me?
Let $E$ be a holomorphic vector bundle with hermitian metric $h$ over
a connected compact Kähler manifold $X$ with Kähler form ...

**2**

votes

**1**answer

159 views

### The compatibility of the Gysin sequence with mixed Hodge structures

Let $X$ be a compact complex $n$-manifold and $D$ be a smooth comdimension $1$ submanifold. Also let $U:= X\setminus D$ and $j$ be the inclusion of $U$ in $X$.
Then it is well known that the ...

**1**

vote

**0**answers

54 views

### Parametric surfaces in $\mathbb{R}^4$ via quaternion multiplication of curves

A curve $r(t)=(u(t), v(t), p(t), q(t)): \mathbb{R} \to \mathbb{R}^4$ can also be thought as a quaternion function $r(t)=u +i v + j p + kq$, where $1,i,j,k$ are the standard basis elements of the ...

**0**

votes

**0**answers

143 views

### Isothermal coordinates

Is there an application or interest in studying the isothermal surfaces where the metric is $ds^2=E∗(du^2+dv^2)$ and where $E>0$ is an harmonic function?
I know that this metric is a special kind ...

**0**

votes

**0**answers

46 views

### Small perturb a continuous map

I am reading the book: Convex Integration Theory by D. Spring and encounter a
question which I subtract as follows.
Let $p: X \rightarrow V$ be a smooth fibre bundle and $\mathcal{R} \subset ...

**4**

votes

**1**answer

162 views

### Hyper-Kaehler Strucutre for Compact Lie Groups?

We know from the classy work of Joyce that "any compact Lie group becomes hypercomplex after it is multiplied by a sufficiently big torus". The quote comes from the Wikipedia page.
I am asking if it ...

**0**

votes

**0**answers

31 views

### A C(B)-module structure on the function algebra of the total space of a vector bunlde $\pi:V \to B$

For a continuous vector bundle $\pi:V \to B$ vector bundle over a compact Hausdorff space $B$, and $C(B)$, $C(V)$ the continuous complex valued functions on $B$ and $V$ respectively, we can give ...

**2**

votes

**2**answers

258 views

### Is it true that all sphere bundles are some double of disk bundle?

Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you ...

**1**

vote

**0**answers

128 views

### Stronger version of Bertini's theorem

In char 0, is there a generalised version of Bertini's theorem that will ensure that for a proper map $f: Y\rightarrow X$ between smooth projective varieties and for every point $x\in X$ we can find ...

**5**

votes

**0**answers

68 views

### In $(\mathbb{R}^4,\omega_{std})$ is positive symplectic area enough to guarantee a pseudoholomorphic disc representative?

I will present my question in the context that I encountered it, although I believe it probably applies in general context.
Consider $\mathbb{R}^4 \cong \mathbb{C}^2$ with the standard symplectic form ...

**7**

votes

**2**answers

252 views

### Surfaces contained in a ball

In this Paper there is a proof that a closed plane curve of length
$L$ and curvature bounded by $K$ can be contained inside a circle of radius
$L/4 - (\pi - 2)/2K$. Are there similar results for ...

**2**

votes

**1**answer

215 views

### Relative tangent bundle and trivilization, tautological foliation

Let $T_{X}\rightarrow X$ be the tangent bundle over a complex manifold $X.$ Let $\pi:PT_{X}\rightarrow X$ be a projectivization of that bundle. Let $L$ be the tautological line bundle of $PT_{X}.$
...

**29**

votes

**5**answers

1k views

### Surfaces filled densely by a geodesic

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no
single geodesic $\gamma$ that fills $S$ densely?
Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points
...

**5**

votes

**3**answers

3k views

### Hessian as a tensor, multi-dimensional taylor series, and generalizations

The Hessian matrix $\{\partial_i \partial_j f \}$ of a function $f:\mathbb{R}^n \to \mathbb{R}$ depends on the coordinate system you choose. If $x_1,\cdots,x_n$ and $y_1,\cdots,y_n$ are two sets of ...