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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
2
votes
0
answers
84
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Green’s function vector bundle laplacian
On a compact Riemann surface with a metric, there exists a Green’s function $C ln(d(x,y)^2)\leq G(x,y)\leq 0$ satisfying $u=\int u+ \int G(x,y) u(y) dy$.
Suppose $(E,h)$ is a Hermitian holomorphic bu …
2
votes
0
answers
114
views
Equivariant resolution of singularities with equivariant centres
From what I understand, given a complex projective variety X inside a compact complex manifold Y, according to Hironaka, there is a sequence of $r$ blowups $Y_i$ of Y along complex submanifolds (centr …
14
votes
1
answer
1k
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When is a given matrix of two forms a curvature form?
Let's assume we are working over $\mathbb{R}^n$ (but feel free to change to domain to answer the question). I wish to know if the equation $F = dA + A \wedge A$ can be solved for a matrix of 1-forms $ …
4
votes
1
answer
215
views
Examples of surfaces with negative Kahler curvature operator
Compact ball quotients are examples of compact Kahler surfaces with negative curvature operator.
Are there any other examples ? What about nonpositive (other than the product of two Riemann surfaces …
4
votes
0
answers
189
views
Chern-Weil theory for coherent subsheaves
If $(E,h)$ is a holomorphic vector bundle over a compact complex manifold $M$ (which is a projective variety), and $S$ is a coherent subsheaf of $M$, then $S$ is secretly a holomorphic vector bundle o …
2
votes
Tubular neighborhoods of embedded manifolds
If you take the ambient manifold to be $\mathbb{C}^n$ with the Euclidean metric and take X as a graph $z_n=f(z_1,..,z_{n-1})$ where f is a polynomial, then it has a uniform tubular neighbourhood. This …
3
votes
What results are immediately generalised to higher dimensions, in light of Schoen and Yau's ...
The compactness of solutions to the Yamabe problem (and its version with boundary) holds upto dimension 24 if the PMT is true. (For higher dimensions there are counterexamples.)
1
vote
0
answers
97
views
Systematic way of finding balanced metrics
In several PDE involving metrics (like the Hermite-Einstein equation for vector bundles and the constant scalar curvature Kahler equation for manifolds) there is a notion along these lines - If a solu …
11
votes
What is a Futaki invariant, what is the intuition behind it, and why is it important?
The Futaki invariant $F(X,[\omega])$ is a quantity that needs two pieces of information on a compact complex manifold $M$.
1) A Kahler class $[\omega]$
2) A holomorphic vector field $X$.
It is an …
2
votes
1
answer
223
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. M …
1
vote
1
answer
203
views
A continuous version of Teichmuller uniqueness
By the Teichmuller uniqueness theorem, given a homeomorphism $f:X \rightarrow X$ where $X$ is the $n$-punctured sphere, there is a unique quasiconformal homeomorphism $g$ fixing $0$, $1$, and $\infty …
12
votes
1
answer
3k
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Density of smooth functions in Sobolev spaces on manifolds
Hebbey defines the Sobolev space of functions on a Riemannian manifold (M,g) as the completion of smooth functions under the Sobolev norm. However, I have seen (elsewhere) that Sobolev spaces have bee …
3
votes
1
answer
275
views
A (non-Kahler) metric on projectivised vector bundles
Given a hermitian holomorphic vector bundle (E, h) on a complex manifold-with-a-metric (X,g), then consider the following (natural) construction of a metric on the total space of $\mathbb{P}(E)$ : Fir …
1
vote
1
answer
304
views
Mathematical software for Chern-Weil theory
I need to compute curvature forms and Chern-Weil forms for a given metric (in local coordinates) on a vector bundle. Is there are software package that does this? If I manage to compute the derivative …
3
votes
1
answer
461
views
Regarding Discrete Eigenvalues
For many eigenvalue problems for differential operators (for example the quantum harmonic oscillator (HO)), unless we impose some behaviour at infinity, the eigenvalues will not be discrete.
But, supp …