**2**

votes

**0**answers

84 views

### Intermediate quotient for a Hermitian Symmetric Spaces of $Sp(n)$

We know that $U(N)$ can be embedded into $SU(n+1)$ and that the quotient is isomorphic to complex projective space:
$$
SU(n+1)/U(n) \simeq {\mathbb CP}^{n}.
$$
We can split this process into two ...

**2**

votes

**1**answer

118 views

### SU(2) invariant Kahler metrics on products of Riemann surfaces

Let $\Sigma$ is a compact Riemann surface with the trivial action of SU(2) and let $\mathbb{P}^1$ be equipped with the standard SU(2) action. Then $X=\Sigma \times \mathbb{P}^1$ has an SU(2) action. ...

**2**

votes

**1**answer

84 views

### Regularity of Hodge Laplacian on bounded domains

I need a reference for the $W^{s,p}$ regularity of the Hodge boundary value problem on bounded domains. I need estimates
$\lVert \omega \rVert_{W^{s+2,p}} \leq c \lVert f \rVert
_{W^{s,p}}$, for ...

**3**

votes

**1**answer

186 views

### Let $X$ be a projective variety and let $Pic(X) \cong \mathbb Z$, then $(X, −K_X) $ is $K$-semistable

Is there any approach for the following conjecture?
Let $X$ be a projective Fano manifold and let $Pic(X) \cong \mathbb Z$, then $(X, −K_X) $ is $K$-semistable.

**4**

votes

**0**answers

126 views

### Is there an example of a holomorphic vector bundle whose Atiyah class vanishes and does not admit a flat connection?

Let $E\to X$ be a holomorphic vector bundle over a Kähler manifold. The vanishing of the Atiyah class, $At(E)=0$, is equivalent to the existence of a holomorphic connection on $E$.
Moreover, it is ...

**12**

votes

**1**answer

134 views

### Smooth manifolds as idempotent splitting completion

The nlab has a particularly interesting thing to say about the category of smooth manifolds: it is the idempotent-splitting completion of the category of open sets of Euclidean spaces and smooth maps.
...

**7**

votes

**1**answer

65 views

### Convergence of functions on Alexandrov spaces

Consider a sequence of $n-$dim Alexandrov spaces with curvature $\geq$ -1 $\{(M_i,p_i)\}$ Gromov-Hausdroff converging to an $n-$dim Alexandrov space $(M,p)$. Let $f:M\mapsto \mathbb R$ be a Lipschitz ...

**0**

votes

**1**answer

100 views

### Local diffeomorphism from a torus to a Lie group

Let $G$ be a simple Lie group of dimension $n$ (connected or even simply connected). Let $T$ be a maximal torus of dimension $d$. Notice that $\frac{n}{d}$ is an integer which I will denote by $m$. ...

**3**

votes

**1**answer

196 views

### To what extent are toric manifolds and principal torus bundles “the same thing”?

I am a little confused by the different definitions for toric manifolds/varieties. Depending on the definition of toric manifolds and principal torus bundles that one chooses, when is a toric manifold ...

**18**

votes

**2**answers

701 views

### how do you visualize characteristic class?

For cohomology, there are some equivalent definitions when the object we consider is sufficiently nice. Since I mainly work with algebraic variety, I will restrict the objects I am considering to be ...

**3**

votes

**1**answer

86 views

### Reference: Finsler Derivative?

On the wikipedia page "Generalizations of derivative" the author mentions: " in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some ...

**5**

votes

**1**answer

112 views

### geodesic of $\rm SO(3)$ as a compact Lie group vs as a Riemannian symmetric space

I got a little bit confused about the definition of geodesic for $\rm SO(3)$ as
a compact Lie group
a Riemannian symmetric space
In the former case, it is given by the usual matrix exponential:
$$
...

**1**

vote

**1**answer

82 views

### Hermitian manifold with harmonic holomorphic volume form

Let M be a compact complex 3-manifold with trivial canonical line bundle and Ω be the non-vanishing holomorphic 3-form.
If the real and imaginary part of Ω are both harmonic with respect to the ...

**5**

votes

**3**answers

347 views

### Determining geodesics between two points in curved space [closed]

In order to determine the geodesics between two points, one must solve the geodesic differential equations, which are as following
\begin{align}
p &= u'(s)\\
q &= v'(s)\\
p' + \Gamma^0_{00}p^2 ...

**1**

vote

**2**answers

183 views

### Geodesic on Banach Manifold [closed]

Is there a way of defining a geodesic on a Banach Manifold $M$ which is not itself a Hilbert Manifold?

**0**

votes

**0**answers

36 views

### Hilbert Manifold structure on global sections of a sheaf

Let $<U_i,\phi_i>$ be an atlas on a Hilbert manifold $M$. If $F(M,U_i)$ is a Hilbert space for every $U_i$ then does this imply that $F(M,M)$ can be made into a Hilbert manifold also with ...

**3**

votes

**0**answers

53 views

### smooth topos as generalized smooth space

I'm interested in generalized smooth spaces. I know there are several spaces such as Deffeological space, Frölicher space, Chen space, etc... and there are some papers compare them. However, I haven't ...

**2**

votes

**1**answer

91 views

### Why simple closed curves are dense in $\mathcal{PML}_0(S)$?

I have another question about laminations on surfaces. As usual let $\mathcal{S}$ be the set of homotopy classes of simple closed curves in $S$ and $\mathcal{PML}_0(S)$ be the set of projective ...

**16**

votes

**4**answers

479 views

### Triangle centers from curve shortening

The curve-shortening flow transforms curves in the plane by moving each point perpendicularly to the curve at a speed proportional to the curvature at that point. It is usually defined for smooth ...

**10**

votes

**1**answer

315 views

### Examples of differential cohomology in cohesive $\infty$ topos

I might direct this question to Urs Schreiber directly, but just in case someone else has some interesting examples, I'll make the question public.
The formulation of differential cohomology in ...

**3**

votes

**1**answer

115 views

### Why is $\mathcal{PML}_0(S)$ compact?

I'm starting to study geodesic laminations on hyperbolic surfaces and in particular I'm focusing my attention on $\mathcal{PML}_0(S)$, the space of projective classes of measured geodesic laminations ...

**8**

votes

**1**answer

298 views

### Multiplicity of Laplace eigenvalues

Disclaimer: This is a very heuristic question and I will be satisfied with heuristic insights, if rigorous and precise answers are not possible.
All the examples of closed surfaces (or higher ...

**2**

votes

**0**answers

103 views

### If G-invariant metric is always Kahler-Einstein

Suppose there is an Hermitian symmetric space of compact type X. It is realized in the following way:
$X\hookrightarrow\mathbb{P}^N$ and equip it with induced Fubini-Study metric g. What's more, the ...

**8**

votes

**3**answers

338 views

### Isometry group of a compact hyperbolic surface

Consider a compact surface $M$ of genus $g \geq 2$ with a metric of constant negative curvature. My question is, is it known under what sorts of sufficient conditions such a metric will have ...

**2**

votes

**2**answers

286 views

### Ricci flow and isometry group

It is known (via Kotschwar's uniqueness of backwards Ricci flows) that the isometry group of a Riemannian metric remains unchanged under the Ricci flow. But, one can easily observe that it can change ...

**9**

votes

**5**answers

446 views

### List of generic properties of Riemannian metrics

I am highly interested in compiling a list of generic properties of Riemannian metrics on a (may be compact) manifold in general, or under "relatively broad" assumptions, like generic properties of ...

**2**

votes

**1**answer

139 views

### Some notational questions regarding tangent vectors

I am not a specialist in differential geometry, so I have some difficulties in finding the right words for the following natural things:
First of all it seems that there is a lot of nonequivalent ...

**4**

votes

**1**answer

78 views

### Gromov-Hausdroff convergence for Alexandrov spaces

Let $\{X_n\}_{n=1}^\infty$ be a sequence of compact Alexandrov spaces (with curvature $\geq k$) converging to (in the sense of Gromov-Hausdroff convergence) an Alexandrov spaces $X$, and ...

**2**

votes

**0**answers

96 views

### Modifying monkey saddles

We get negative Gauss curvature K surfaces of quasi polar symmetry surface form generated from:
$$ Re (x+ i y)^n = a^n $$ (n integer) with n humps above plane $ z =0$.
($ n =2,3,4 $ hyperbolic ...

**3**

votes

**0**answers

40 views

### Laplacian Spectra on Nearly Nodal Riemann Surfaces

Consider a family of complex curves ${\mathcal C} \to {\mathbb D}$ such that the central fibre is a nodal Riemann surface while other fibres are smooth Riemann surfaces. We choose a family of ...

**1**

vote

**0**answers

76 views

### Mean value operator on Riemannian manifold

Let $(M,g)$ a Riemannian manifold. Further $M$ should be a harmonic space, that is $M$ is a symmetric and simply connected space of rank 1. (Example: Spheres $S^n$)
Consider the mean value operator, ...

**8**

votes

**0**answers

113 views

### Optimal exponent in the Lojasiewicz-Simon gradient inequality

Lojasiewicz's theorem asserts that if $F: \mathbb{R}^n\to \mathbb{R}$ is a real-analytic function in a neighborhood of its critical point $0$, then there exist constants $\theta\in (0,1/2]$, ...

**6**

votes

**1**answer

113 views

### Self-avoiding/reflecting geodesics on a convex surface

Let $S$ be the surface of a convex body embedded in $\mathbb{R}^3$.
For me $S$ is a convex polyhedron,
but I am happy to view $S$ as a smooth body with positive Gaussian curvature
at each point, or ...

**5**

votes

**2**answers

212 views

### Noninvariance for a specific nonlinear oscillator

Consider the nonlinear system
\begin{align*}
\frac{d}{dt} \begin{pmatrix} x_1(t) \\ x_2(t) \end{pmatrix} = \begin{pmatrix} x_2(t) \\ -4x_1(t) + x_1^2(t) \end{pmatrix},
\end{align*}
which admits ...

**7**

votes

**2**answers

368 views

### What is the obstruction to the existence of a global Kahler potential?

It is a well-know fact that if $(X,\omega)$ is Kahler then about every point $x \in X$ there exists a neighbourhood $U$ and a function $K \in C^{\infty}(U,\mathbb{R})$ such that $\omega|_U = i\partial ...

**4**

votes

**0**answers

92 views

### Alexandrov-Fenchel inequality for sets of positive reach

If $E$ is a convex subset of $\mathbb{R}^n$ with $|E| = |B_1|$, then one consequence of the classical Alexandrov-Fenchel inequalities from convex geometry is that
$$\int_{\partial E} H_{\partial E} ...

**3**

votes

**0**answers

88 views

### Equations of the form $((\Delta)^{2}+\lambda\Delta+\gamma)(f)=0$

Here is the problem: I am trying to figure out if there is a non-zero solution for the system of equations:
$(\Delta^{k-1}+a)(d^{\ast}\eta)=2d^{\ast}d\xi$
$(\Delta^{k}+b)(d\xi)=2dd^{\ast}\eta$
for ...

**4**

votes

**2**answers

156 views

### Unique Kahler-Einstein metric $g$ with $\mathrm{Ricc}(g)=-g$ when first Chern class $C_1(M)<0$: $\mathrm{Ricc}(h)=-g\,\Rightarrow\,h=cg$ for $c>0$?

On a compact Kahler manifold, let $g$ be the unique Kahler-Einstein metric with $\mathrm{Ricc}(g)=-g$, proved to exist by Yau and Aubin when the first Chern class $C_1(M)<0$.
Question: Does $g$ ...

**7**

votes

**0**answers

219 views

### A question on a result of Colin de Verdiere

Consider a compact connected surface $M$ of some genus $\gamma \geq 2$. A particular case of a famous result of Colin de Verdiere (see here) says that if we fix $\gamma$ and select a finite sequence ...

**0**

votes

**1**answer

93 views

### Eikonal equation and double null coordinates

I"m trying to understand the exact/technical link between the Eikonal equation and a double-null form of the metric (if such a direct link even exists). R. Wald, in his "General Relativity", doesn't ...

**1**

vote

**0**answers

40 views

### Attainability of Global Optima In Optimal Control

Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation:
$\frac{d x(t)}{dt} = F(x,u)$
one can consider the ...

**3**

votes

**0**answers

91 views

### Faster (than normal) convergence of the normalized Ricci flow on surfaces

Consider a compact surface $M$ of genus $\gamma > 1$ (I am using the more usual letter "$g$" to denote metric), and the normalized Ricci flow on it. It is known that at time $t$, the scalar ...

**1**

vote

**0**answers

19 views

### The jump set of $SBV$ function over a hyper surface

Assume $\Omega\subset \mathbb R^N$ is open bounded, smooth boundary. Also assume $S\subset \Omega$ is a smooth hyper surface such that $0<\mathcal H^{N-1}(S)<+\infty$.
Now, given a positive ...

**3**

votes

**0**answers

84 views

### What subdomains of $\mathbb{R}^2$ are diffeomorphic to $\mathbb{R}^2_+$ via rational functions?

For what subdomains $D \subset \mathbb{R}^2$ does there exist a rational diffeomorphism $h:D \to \mathbb{R}^2_+$, such that its inverse is also a rational function? (By "rational function" I mean a ...

**10**

votes

**0**answers

149 views

### Can we find minimal-diameter metrics without computability?

A beautiful argument by Nabutovsky and Weinberger (see http://math.uchicago.edu/~shmuel/fractal.ps) shows that, if $M$ is any smooth compact manifold of dimension $\ge 5$, then the diameter functional ...

**2**

votes

**1**answer

151 views

### comparing Laplacian and gradient of function on boundary

Consider $ E(x)$ some smooth function on $ \Omega$ (some smooth bounded domain in $ R^N$) and suppose $E=0$ on $ \partial \Omega$.
Suppose one knows that there is some $C_1,C_2 \in R$ such that
$ x ...

**7**

votes

**0**answers

143 views

### Gauss curvature flow

Let me first start by stating the following wonderfull theorem of Ben Andrews "Gauss curvature flow: the fate of the rolling stones" 1999.
Let $M_0=x_0(M)$ be a compact, smooth, strictly convex ...

**16**

votes

**1**answer

700 views

### When is curvature given by a connection?

Suppose we have a curvature-like tensor $R\in \wedge^2 T^*_p M \otimes T^*_p M \otimes T_p M$ on a manifold $M$, that is $R(X,Y)Z = - R(Y,X)Z$. How does one determine whether or not this is a ...

**5**

votes

**1**answer

234 views

### harmonic extension of a curve by different parametrization

Let us consider a curve $\gamma :S^1 \rightarrow \mathbb{R}^3$ (or even a planar convex one if it simplifies). Then I look to the harmonic extension to the disc $h:\mathbb{D}\rightarrow \mathbb{R}^3$ ...

**3**

votes

**0**answers

40 views

### Foliation by Umbilic surfaces

Suppose $(M,g)$ denotes a Riemannian manifold with boundary that is a foliation by Umbilic surfaces. (As an example consider a manifold where the exists a unit parallel vector field) .
Is it ...