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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 …

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 …

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 $ …

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 …

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 …

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 …

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.)

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …