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Consider $(S^1 \times \Sigma^2, g)$, where $g$ is any Riemannian metric on the compact and closed $3$-manifold $S^1 \times \Sigma^2$. Question: Does there always exist a nowhere vanishing harmonic $1 …

Let $P$ be a principal $G$ bundle. Let $S$ be a space with left action of $G$, and let $Q$ be a principal $H$ bundle over $S$ with the property that the action of $G$ can be lifted to $Q$. Then $$ P \ …

What is known about the moduli space of stable rank $3$ bundles on the projective plane $\mathbb{CP}^2$? Ideally, there is a concrete complex manifold which is a fine moduli space for such bundles for …

Let $V$, $W$ be vector bundles over a compact Riemannian manifold $M$ and let $F$ be a smooth elliptic operator of order $k$ from $V$ to $W$. "Standard elliptic theory" then gives us the following two …

The closest I found: In Michael Plum: Explicit $H_2$ Estimates and Pointwise Bounds for Solutions of Second-Order Elliptic Boundary Value Problems the following explicit estimate is given (equation 3) …

Let $f,g \in \mathbb{R}[x_0,\dots,x_k]$ be homogeneous polynomials and $X:=Z(f) \subset \mathbb{RP}^k$ be the projective variety defined by $f$. Assume that $X$ is smooth and has codimension $1$. Then …