Under broad hypotheses we derive a scalar reduction of the generalized Kähler–Ricci soliton system. We realize solutions as critical points of a functional, analogous to the classical Aubin energy, defined on an orbit of the natural Hamiltonian action of diffeomorphisms, thought of as a generalized Kähler class. This functional is convex on a large set of paths in this space, and using this we show rigidity of solitons in their generalized Kähler class. As an application we prove uniqueness of the generalized Kähler–Ricci solitons on Hopf surfaces constructed in Streets and Ustinovskiy [Commun. Pure Appl. Math. 74(9), 1896–1914 (2020)], finishing the classification in complex dimension 2.

In this paper, we prove a volume growth estimate for steady gradient Ricci solitons with bounded Nash entropy. We show that such a steady gradient Ricci soliton has volume growth rate no smaller than $r^{\frac{n+1}{2}}.$ This result not only improves the estimate in (Chan et al., arXiv:2107.01419, 2021, Theorem 1.3), but also is optimal since the Bryant soliton and Appleton’s solitons (Appleton, arXiv:1708.00161, 2017) have exactly this growth rate.

We construct (modified) scattering operators for the Vlasov–Poisson system in three dimensions, mapping small asymptotic dynamics as $t\rightarrow -\infty$ to asymptotic dynamics as $t\rightarrow +\infty$. The main novelty is the construction of modified wave operators, but we also obtain a new simple proof of modified scattering. Our analysis is guided by the Hamiltonian structure of the Vlasov–Poisson system. Via a pseudo-conformal inversion, we recast the question of asymptotic behavior in terms of local in time dynamics of a new equation with singular coefficients which is approximately integrated using a generating function.

In this article, we present the concavity of the minimal $L^2$ integrals related to multiplier ideals sheaves on Stein manifolds. As applications, we obtain a necessary condition for the concavity degenerating to linearity, a characterization for 1-dimensional case, and a characterization for the equality in 1-dimensional optimal $L^{2}$ extension problem to hold.

In this note we derive an improved no-local-collapsing theorem of Ricci flow under the scalar curvature bound condition along the worldline of the basepoint. It is a refinement of Perelman’s no-local-collapsing theorem.

A KdV flow is constructed on a space whose structure is described in terms of the spectrum of the underlying Schrödinger operators. The space includes the conventional decaying functions and ergodic ones. Especially, any smooth almost periodic function can be initial data for the KdV equation.

Let G be a connected, complex reductive group. In this paper, we classify $G\times G$-equivariant normal ${\mathbb {R}}$-test configurations of a polarized G-compactification. Then, for ${\mathbb {Q}}$-Fano G-compactifications, we express the H-invariants of their equivariant normal ${\mathbb {R}}$-test configurations in terms of the combinatory data. Based on Han and Li “Algebraic uniqueness of Kähler–Ricci flow limits and optimal degenerations of Fano varieties”, we compute the semistable limit of a K-unstable Fano G-compactification. As an application, we show that for the two smooth K-unstable Fano SO$_4({\mathbb {C}})$-compactifications, the corresponding semistable limits are indeed the limit spaces of the normalized Kähler–Ricci flow.

In this paper, we classify Möbius invariant differential operators of second order in two-dimensional Euclidean space, and establish a Liouville type theorem for general Möbius invariant elliptic equations. The equations are naturally associated with a continuous family of convex cones $\Gamma _p$ in $\mathbb R^2$, with parameter $p\in [1, 2]$, joining the half plane $\Gamma _1:=\{ (\lambda _1, \lambda _2):\lambda _1+\lambda _2>0\}$ and the first quadrant $\Gamma _2:=\{ (\lambda _1, \lambda _2):\lambda _1, \lambda _2>0\}$. Chen and C. M. Li established in 1991 a Liouville type theorem corresponding to $\Gamma _1$ under an integrability assumption on the solution. The uniqueness result does not hold without this assumption. The Liouville type theorem we establish in this paper for $\Gamma _p$, $1<p\le 2$, does not require any additional assumption on the solution as for $\Gamma _1$. This is reminiscent of the Liouville type theorems in dimensions $n\ge 3$ established by Caffarelli, Gidas and Spruck in 1989 and by A. B. Li and Y. Y. Li in 2003–2005, where no additional assumption was needed either. On the other hand, there is a striking new phenomena in dimension $n=2$ that $\Gamma _p$ for $p=1$ is a sharp dividing line for such uniqueness result to hold without any further assumption on the solution. In dimensions $n\ge 3$, there is no such dividing line.

Using an argument of Baldwin–Hu–Sivek, we prove that if K is a hyperbolic fibered knot with fiber F in a closed, oriented 3-manifold Y, and $\widehat{HFK}(Y,K,[F], g(F)-1)$ has rank 1, then the monodromy of K is freely isotopic to a pseudo-Anosov map with no fixed points. In particular, this shows that the monodromy of a hyperbolic L-space knot is freely isotopic to a map with no fixed points.