Search Results

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 $ …

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 …

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 …

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 …

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.)

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …