# Uniqueness of singular Hermitian-Einstein metric along Yau-Donaldson flow?

The following question is related to Singular Yang-Mills theory

The Hermitian metric $H$ along the fibers of the holomorphic vector bundle $E$ over a K\"ahler manifold $(M,\omega)$ is Hermitian-Einstein if $$\sqrt[]{-1}\Lambda_\omega F_H = \gamma I$$ at every point of $M$, where $\gamma$ is a global constant and $I$ is the identity endomorphism of $E$ and $F_H$ is the Chern curvature form of $(E, h)$. Here $\Lambda_\omega$ is the contraction with $\omega$ which means $$F_H\wedge\omega^{n-1}=(n-1)!\Lambda_\omega F_H\frac{\omega^n}{n!}$$

If a vector bundle $E$ admit Hermitian-Einstein metric, then the trace free part of $F_H^o=F_H-(\frac{1}{rk E}Tr F_H)$ is harmonic form.

Now take the integral $$D_\lambda(H_0,H)=\int_M(R_2(H,H_0)-2\lambda R_1(H,H_0)\wedge \omega)\wedge \frac{\omega^{n-1}}{n!}$$ where $\bar\partial\partial R_2(H,H_0)=4\pi^2(\pi_1(E,H)-\pi_1(E,H_0))$ and $\bar\partial\partial R_1(H_1,H_0)=2\pi(c_1(E,H)-c_1(E,H_0))$ where $\pi_1(E,H_0)$ is the first Pontryagin form.

Now, lets define the Donaldson flow (which is like Kahler-Ricci flow version(due to Yau) for Vector bundles) .

Let $\mathcal M$ be the Frechet space of hermitian metrics on $E$, we define $\theta$ as follows $\mathcal X\in T_H\mathcal M$, $$\mathcal X(u,v)=H(\theta(\mathcal X)u,v)$$

Now Take $H_t=H_0.h_t$ then $$\frac{\partial H_t}{\partial t}=-2\theta^{-1}(\Lambda F_{H_t}-\lambda Id)$$

can be written as follows

$$\frac{\partial h_t}{\partial t}=-2h_t(\Lambda F_{h_t}-\lambda Id)$$

called Yau-Donaldson flow

Hence we are facing with a nonlinear PDE strictly parabolic since we can write it as follows

$$\frac{\partial h_t}{\partial t}=-\Delta h_t-\{\Lambda_{H_0}-\lambda Id,h\}+2i\Lambda\bar\partial h_t.h_t^{-1}\wedge \partial_0 h_t$$

where $\{,\}$ represent anti-commutator of $End(E)$. In fact the existence of the solutions of such flow correspond to local inverse theorem of Banach in smooth setting, but for singular setting, I don't know(since perturbation method does not work).

Now if we choose the Initial metric as smooth Hermitian metric and such flow admit a solution $H_\infty$ called Hermitian-Einstein metric then for all other Hermitian metric $H$ we have $D_\lambda(H,H_\infty)>0$, this tells us that such flow admit at most one solution and we get unicity of the solution of Hermitian-Einstein metric. But for singular Hermitian-Einstein metric in general we don't have $D_\lambda(H,H_\infty)>0$ and it can be negative.

Or in smooth case if we set

$\tau (h1; h2) = Tr(h_1^{-1}h_2)$ $\sigma(h_1; h_2) = \tau (h_1; h_2) + \tau (h_2; h_1) - 2 rkE$

that $\sigma (h_1; h_2) \geq 0$ with equality if and only if $h_1 = h_2$.

Now if $h_1(t)$ and $h_2(t)$ are two solutions of the Hermitian-Yang Mills equation: $$\frac{\partial h_t}{\partial t}=-2h_t(\Lambda F_{h_t}-\lambda Id)$$ then if we write $\sigma = \sigma(h_1(t); h_2(t))$ then we have

$$\frac{\partial \sigma}{\partial t}+\Delta \sigma\leq 0$$ and we get by maximum principle the uniqueness of the solution, but if $h_1(t)$ and $h_2(t)$ are singular then $\sigma$ is singular and we can not use maximum principle in general.

So we need to add some condition on the initial singular Hermitian metric to run the Yau-Donaldson flow and get uniqueness. Something like vanishing Lelong number. Which information on the initial metric is needed to get the unique solution for Yau-Donaldson flow?(Here I mean well defined singular Hermitian metric, since in general such metric is not well defined)

As a motivation, when we consider parabolic stability for framed Vector bundle on pair with a snc divisor $(X,D)$ we need to add some condition on initial metric such that to control blowing up along divisor and get uniqueness of the solutions . Recently a lot of papers published like mushroom without considering this important point. As an example Existence of Hermitian-Einstein metric on stable Orbifold bundle on orbifolds which initial metric can be non-smooth in general. Or finding twisted Hermitian-Einstein metric on holomorphic fiber space needed to assume initial metric to be singular, see my question in MO

Or from the work of Y.T. Siu for existence of Hermitian-Einstein metric on reflexive sheaf

Let $(X,\omega)$ be a compact Kahler manifold and $\mathcal E$ be a reflexive sheaf on $X$. Then there exists a resolution $0\to \mathcal E\to E_0\to E_1\to E_2\to\cdots$ of vector bundles $E_i$ and if we take a hermitian metric on vector bundle $E_0$ then it induces a singular hermitian metric which is well defined on $\mathcal E$ on $X$ outside of closed subset $S$ of codim 4. and then they start a heat equation with such initial hermitian metric which is not smooth in general and they didn't proof the uniqueness of solutions and took it as trivial fact

It is well known that the uniqueness of a long time solution to the Hermitian-Yang-Mills flow implies the uniqueness of a long time solution to the Yang-Mills flow. But it does not work for singular connection

• It would probably be a good idea to emphasize your actual question. – Mariano Suárez-Álvarez Apr 30 '17 at 3:24
• @MarianoSuárez-Álvarez , I this question, I have considered just uniqueness of such solutions. – user21574 Apr 30 '17 at 3:45