What is μ-cscK metric? (in writing)

I will add and polish explanations when I feel like it.

  Constant μ-scalar curvature Kähler metric (μ-cscK metric) is a framework unifying cscK metric and Kähler-Ricci soliton, which I proposed in mu-cscK paper. A $\mu^\lambda_\xi$-cscK metric is a Kähler metric $g$ satisfying the equation $s (g) + \Delta \mu_\xi - \frac{1}{2} |\nabla \mu_\xi|^2 = \lambda \mu_\xi$. There are two motivative observations for this concept.

  Moment map picture: The first observation is Donaldson-Fujiki type moment map picture for Kähler-Ricci soliton. When there is a Hamiltonian torus action on a symplectic manifold $(M, \omega)$, we can consider a symplectic structure $\Omega_\chi$ on the space $\mathcal{J}_T (M, \omega)$ of torus invariant almost complex structures, associated to each weight function χ on the moment polytope $P = \mu (M)$. A moment map associated to this symplectic structure is expressed by weighted scalar curvature introduced by Lahdili (cf. weighted cscK metric (Lahdili)). For a symplectic manifold $(M, \omega)$ satisfying $\lambda [\omega] = 2\pi c_1 (M, \omega)$ for $\lambda > 0$, exponential weight $\chi = \exp (\langle \xi, \cdot \rangle)$ gives a moment map picture for Kähler-Ricci soliton $\mathrm{Ric} (g) + L_{J\xi/2} g = \lambda g$ (cf. moduli paper). The μ-scalar curvature $s_\xi (g) = s (g) + \Delta \mu_\xi - \frac{1}{2} |\nabla \mu_\xi|$ is equivalent to the weighted scalar curvature associated to the exponential weights.
  As a general fact, moment maps are unique modulo the center of the group action. Since $\mathfrak{t} = \mathrm{Lie} (T)$ is the center of $\mathrm{Lie} (\mathrm{Ham}_T (M, \omega))$, the translation of the moment map $J \mapsto s_\xi (g) e^{\mu_\xi} \omega^n$ by a constant $\mu_\zeta e^{\mu_\xi} \omega^n \in \mathrm{Lie} (\mathrm{Ham}_T (M, \omega))^\vee$ gives a new moment map.
  We are interested in the zero set of these moment maps. It is characterized by the equation $s_\xi (g) = \mu_\zeta$ for some $\xi, \zeta \in \mathfrak{t}$. By a general argument on moment map, we observe $\mathrm{Fut}_\xi := \int_X \mu_\bullet s_\xi (g) e^{\mu_\xi} \omega^n, |\zeta \rangle_\xi := \int_X \mu_\bullet \mu_\zeta e^{\mu_\xi} \omega^n \in \mathfrak{t}^\vee$ is independent of $J \in \mathcal{J}_T(M, \omega)$. These invariants give an obstruction for the existence of metrics satisfying $s_\xi (g) = \mu_\zeta$: the zero set of the moment map is non-empty only if $\mathrm{Fut}_\xi = |\zeta \rangle_\xi$. Since the map $| \cdot \rangle_\xi: \mathfrak{t} \to \mathfrak{t}^\vee$ gives a non-degenerate pairing, there is unique $\zeta \in \mathfrak{t}$ satisfying $\mathrm{Fut}_\xi = |\zeta \rangle_\xi$ for each $\xi$. At this point, there is no constraint on $\xi$.
  Now let us recall how Kähler-Ricci soliton is interpreted in this framework. Kähler-Ricci soliton $\mathrm{Ric} (g) + L_{J\xi/2} g = \lambda g$ is equivalent to the equation $s_\xi (g) = \lambda \mu_\xi$ for $g$ in the anti-canonical class $2\pi \lambda^{-1} c_1 (X)$ of a Fano manifold $X$. In order to make our framework equivalent to Kähler-Ricci soliton for Kähler metrics in the Kähler class $2\pi \lambda^{-1} c_1 (X)$, we must impose $\zeta = \lambda \xi$ for $\zeta$ satisfying $\mathrm{Fut}_\xi = |\zeta \rangle_\xi$. This turns into a constraint $\mathrm{Fut}_\xi = \lambda |\xi \rangle_\xi$ on $\xi$. As we explain below, this constraint characterizes $\xi$ uniquely, which is nice in view of moduli theory.
  We want to consider a generalization of this situation for general polarization. Let us impose some dependency on $\zeta$, say $\zeta = \Delta (\xi)$ by some $\Delta: \mathfrak{t} \to \mathfrak{t}$. We call a Kähler metric $g$ is a $\mu^\Delta_\xi$-cscK metric if it satisfies the equation $s_\xi (g) = \mu_{\Delta (\xi)}$. For instance, we can take $\Delta (\xi) = 0$, $\Delta (\xi) = \lambda \xi$ or $\Delta (\xi) = | \cdot \rangle_\xi^{-1} \circ \mathrm{Fut}_\xi$. In this situation, we get a constraint $\mathrm{Fut}_\xi = |\Delta (\xi) \rangle_\xi$ on $\xi$. (As for the last $\Delta$ in the example, it does not impose any constraint on $\xi$. This is not interesting in view of volume minimization. )
  Among many possible choice, scaling dependence $\Delta (\xi) = \lambda \xi$ seems the simplest choice enclosing the theory on Kahler-Ricci solitons. I decided to begin with this simplest case, regarding the $\lambda$ as a free parameter in the theory. This seems a nice decision: we can generalize Tian-Zhu's volume minimization argument, can deepen the corresponding K-stability based on the equivariant intersection formula and, especially, can find a deep connection with Perelman's entropy and can formulate optimal destabilization problem. We also encounter many interesting phenomenon: extremal limit $\lambda \to -\infty$, phase transition phenomenon for $\lambda \gg 0$.

  Volume minimization: Moment map picture shows the well-definedness and the $T$-equivariant deformation invariance of the $\mu$-Futaki invariant $\mathrm{Fut}^\lambda_\xi \in \mathfrak{t}^\vee$. It vanishes when the Kähler class admits a $\mu^\lambda_\xi$-cscK metric. Tian-Zhu's volume minimization argument shows the unique existence of $\xi$ with vanishing modified Futaki invariant $\mathrm{Fut}_\xi^{2\pi} = 0$ for any Fano manifold, regardless of the existence of Kähler-Ricci soliton. In mu-cscK paper, I studied the existence and the uniqueness of $\xi$ with $\mathrm{Fut}^\lambda_\xi = 0$ for general $\lambda \in \mathbb{R}$ and general polarization $L = [\omega]$. Before explaining the results, I would like to share my personal belief on the importance of Tian-Zhu's volume minimization argument.
  Firstly, it is important in moduli theory on Kähler-Ricci solitons. Algebro-geometric moduli problem is a problem to find an appropriate definition of moduli stack and to construct an algebraic moduli space which is characterized by a good geometric relation with the moduli stack: it has a universal morphism from the moduli stack and the morphism enjoys a some geometric conditions. In moduli paper, we consider the stack consisting of $T$-equivariant families of Fano manifolds which are modified K-semistable with respect to the vector $\xi \in \mathfrak{t}$ determined by Tian-Zhu's argument. It admits a moduli space parametrizing the isomorphism classes of Fano manifolds which is modified K-polystable, rather than the isomorphism classes of pairs $(X, \xi)$, as the optimal vector $\xi$ is recovered from the $T$-action by Tian-Zhu's argument (and as the $T$-actions are conjugate if $X \cong X'$). Restricting to $T$-equivariant families is crucial for the separatedness of the moduli space.
  Secondly, volume minimization is an appearance of optimal destabilization.
I will write more on this later, comparing it with other frameworks such as normalized Donaldson-Futaki invariant, normalized volume, H-entropy....
  The main conclusion of the volume minimization argument is that for each $\lambda \in \mathbb{R}$, there always exists $\xi$ satisfying $\mathrm{Fut}^\lambda_\xi = 0$. Studying the limit $\lambda \to - \infty$, we can show such $\xi$ is unique when $\lambda \ll 0$. Combining this with Lahdili's convexity result on weighted Mabuchi functional (cf. convexity of weighted Mabuchi (Lahdili)), we obtain the uniqueness of $\mu^\lambda$-cscK metrics for $\lambda \ll 0$. I conjecture the uniqueness holds as long as $\lambda \le 0$. When $\lambda \gg 0$, we face with totally different phenomenon as we explain below.

  Phase transition The μ-cscK metric possesses a natural parameter $\lambda$ of freedom. There are interesting phenomenons that we are motivated to interpret the parameter as `temperature'.
  Firstly, extremal metric appears in the limit $\lambda \to -\infty$ of $\mu^\lambda$-cscK metrics. We may regard the parameter as a continuity path connecting Kähler-Ricci soliton and extremal metric on Fano manifolds, or connecting the simplest $\mu^0$-cscK metric and extremal metric on general polarized manifold.
  Contrary to the case $\lambda \ll 0$, studying the limit $\lambda \to + \infty$, it turns out that $\xi$ satisfying $\mathrm{Fut}^\lambda_\xi = 0$ are never unique for $\lambda \gg 0$. Indeed, we can construct a $\mu^\lambda$-cscK metric on $\mathbb{C}P^1$ which is not a cscK metric for $\lambda \gg 0$. The isometry of the new $\mu$-cscK metric reduces to $U(1)$. This phenomenon reminds me phase transition and symmetry breaking in physics. It must be interesting if we can give a statistical mechanical interpretion to this phenomenon.
I will write more on this later.



  μK-stability: Since we have a moment map picture for μ-cscK metrics, it is natural to believe there is a notion of stability of polarized variety which encloses K-stability and modified K-stability. I will write more on this later. See for instance weighted cscK metric (Lahdili) by Abdellah Lahdili and muK-stability paper as references.



  Perelman's entropies
I will write this later. See mu-entropy paper for instance.

  Non-archimedean μ-entropy
I will write this when I finish writing my second paper on mu-entropy.