Papers

We prove an invariant identity relating the curvature operators of the first and second kind in dimension four. Under the natural identification $S^2_0\cong\Lambda^+\otimes\Lambda^-$, the trace-free Ricci tensor contributes through a single commutator term. As a consequence, we relate the curvature operator of the second kind on products of unit self-dual and anti-self-dual two-forms to biorthogonal sectional curvature. We then classify closed oriented four-manifolds satisfying $K^\perp_{\max}\leq S/6$, where $K^\perp_{\max}$ denotes the maximum biorthogonal sectional curvature and $S$ is the scalar curvature. The classification includes the cases in which the scalar curvature vanishes.

For an oriented regular plane curve $\gamma$, define ${\cal T}\gamma=\gamma+\tau$, where $\tau$ is its unit tangent. We construct a noncircular curve $\Gamma_0$ such that every iterate ${\cal T}^n\Gamma_0$, $n\geq0$, is a smooth embedded closed curve with everywhere positive curvature. This answers a question of Tabachnikov. The construction starts with a noncompact convex curve $C$, asymptotic at its two ends to parallel lines, satisfying ${\cal T} C=C+(V,0)$ for some $V>0$. From $C$ we construct long closed convex curves. After matching perimeters, we compare each curve with a convex curve whose unit-tangent image is the next curve in the sequence. In suitable arclength coordinates, their curvature functions differ by an exponentially small amount in $L^1$. Uniform inverse estimates turn these approximate relations into an exact sequence $\Gamma_{n+1}={\cal T}\Gamma_n$. Each $\Gamma_n$ lies in a strip of width bounded independently of $n$, whereas the radii of iterated circles tend to infinity.

Under the assumption of integrability of the smooth Jacobi fields and a condition on the intersection pattern of the link, we prove uniqueness of special Lagrangian tangent cones in $\mathbb{C}^3$ whose links are compact immersed special Legendrian surfaces with transverse double points. The proof is based on a discrete Lojasiewicz type inequality, which comes from a gluing obstruction mechanism.

We develop a modular wall crossing theory for quasimaps to a projective variety, allowing independent variation of the boundary coefficients and the quasimap weight. Building on the K-stability of quasimaps introduced by Hashizume and the first author, we construct projective moduli spaces in the stable, Calabi–Yau, and log Fano regimes, together with wall crossing morphisms. A central construction is the moduli theory of boundary polarized Calabi–Yau quasimaps, which retains an ample polarization at the numerically trivial locus and allows comparison with suitable perturbations toward the stable and log Fano regions. The stable theory applies in arbitrary genus, while the comparisons through the Calabi–Yau locus concern genus zero. For degree-one boundary divisors, the resulting framework relates weighted stable maps and quasimaps to Hassett spaces and GIT quotients of weighted points on $\mathbb P^1$. In a companion paper, we apply this framework to give a modular interpolation between Miranda's GIT compactification of rational elliptic surfaces and the Baily–Borel compactification of an eight-dimensional ball quotient.

Using K-moduli spaces for log quasimaps $q_t\colon\left(\mathbb P^1,\frac{1-t}{12}D\right)\to[\mathbb A^2/\mathbb G_m]$ of degree twelve with twelve points and weight $t/12$, we construct a modular interpolation $\{\mathcal M_t\}_{0\le t\le1}$ between the Baily–Borel compactification of the Heckman–Looijenga ball quotient $X_o$ and Miranda's GIT compactification of the moduli space of rational elliptic surfaces. We completely determine the wall-crossing and, for every rational $t\in[0,1]$, identify \[ \mathcal M_t\cong\operatorname{Proj}R\!\left(X_o,\mathcal L+\frac t2\Delta(6)+\frac t3\Delta(9)\right), \] where $\mathcal L$ is the automorphic $\mathbb Q$-line bundle and $\Delta(6),\Delta(9)$ are distinguished Heegner divisors. On the automorphic side, we construct a new automorphic form on $X_o$ via a Borcherds product, whose divisor gives an independent relation among the Heegner divisors. This relation provides a key input for determining the birational transformations in the K-moduli wall-crossing. As part of this analysis, we show that the first positive chamber $\mathcal M_t$ for $t\in (0,1/7)$ is Looijenga's semi-toroidal compactification.

In a neighborhood of the origin in $\mathbb{R}^9$, we construct a minimal hypersurface with respect to a smooth conformally Euclidean metric, whose singular set consists of the origin and a sequence of isolated singular points converging to it. At each of the isolated singular points the hypersurface has the unique tangent cone $C_{4,3}$, the cone over $S^4(\sqrt{4/7})\times S^3(\sqrt{3/7})$, whereas at the origin its unique tangent cone is the cone $C_{3,3}\times\mathbb{R}$. The metric is Euclidean near each isolated singular point and agrees with the Euclidean metric to infinite order at the origin. A key step, of independent interest, is the construction of an entire stationary hypersurface in Euclidean $\mathbb{R}^9$ with exactly one singular point at the origin, at which its unique tangent cone is $C_{4,3}$, and whose unique tangent cone at infinity is $C_{3,3}\times\mathbb{R}$.

A $\lambda$-self-expander $F: M^n\to \mathbb{R}^{n+1}$ is the solution of the isoperimetric problem of weight $e^{\frac{|x|^2}{4}}$. In this paper, for sufficiently large $\lambda$, we construct an immersed non-embedded spherical $\lambda$-self-expander in $\mathbb{R}^{n+1}$ with the mean curvature $H>0$. Moreover, we obtain the explicit solution for $n=2$.

We determine the sharp compactness ranges for the scalar-flat and minimal-boundary Yamabe problems on smooth compact manifolds of positive conformal type, excluding the conformal round hemisphere. For zero scalar curvature and positive constant boundary mean curvature, compactness holds through dimension $14$ for general boundary and dimension $21$ for umbilic boundary. For positive scalar curvature and zero boundary mean curvature, the corresponding upper dimensions are $14$ and $20$. Together with the noncompactness examples in Part I, these results identify the transition dimensions in both boundary classes. For positive scalar curvature, we also prove compactness through dimension eight for every fixed real boundary mean curvature, and obtain higher-dimensional ranges when this curvature is near zero or sufficiently large and positive. The proof combines scalar-correction estimates for the full conformal Fermi metric expansion with a geometric formula expressing the logarithmic coefficient of the corrected energy as a negative sum of squares.

In this paper, we establish a sharp dynamical isoperimetric principle for weak mean curvature flow. We prove that, among weak mean curvature flows with fixed area of initial smooth closed hypersurfaces, the standard smoothly shrinking spherical mean curvature flow uniquely maximizes the extinction time. More precisely, for the level set flow $K_t$ starting from the boundary of a smooth bounded domain $\Omega\subset\mathbb R^{n+1}$, as well as for the integral Brakke flow $\{\mu_t\}$ with initial Radon measure $\mu_0=\mathcal H^n\llcorner\partial\Omega$, we establish the corresponding optimal extinction time estimates \begin{equation*} T_ext(\Omega),\, T^B_ext(\Omega) \leq \frac{1}{2n} \left(\frac{P( \Omega)}{|{\mathbb{S}^n}|}\right)^{\frac{2}{n}}, \end{equation*} where $P(\Omega)$ is the perimeter of $\Omega$ representing the area of $\partial \Omega$, and the equality holds if and only if $\Omega$ is a round ball and the flow is the standard multiplicity-one smoothly self-shrinking round sphere. In particular, we obtain the sharp $L^p$-estimates for the arrival time function of a smooth bounded mean convex domain. In addition, we also establish the sharp isoperimetric inequality for the parabolic measure of the space-time track filling $X$ of outward minimizing level set flow starting from the boundary of a smooth bounded domain $\Omega\subset \mathbb R^{n+1}$: \begin{equation*} \mathcal H_{\mathrm{par}}^{n+2}(X) \leq \frac{\pi} {2n(n+2)^2|{\mathbb{S}^n}|^{{\frac{2}{n}}}} P(\Omega)^{\frac{n+2}{n}}, \end{equation*} where the equality holds if and only if $\Omega$ is a round ball and the flow is standard smoothly shrinking round sphere.

We construct complex structures on the six-sphere $S^6$ with arbitrarily large local Kuranishi dimension. As a corollary, we find the same phenomenon holds for the smooth manifold underlying complex projective 3-space $\mathbb{P}^3$.

This is a continuation of our previous paper on toric gravitational instantons. We develop a systematic theory of compact simply-connected Einstein four-manifolds with toric symmetry. Applications include (1) The construction of simply connected positive Einstein four-manifolds with arbitrarily large $b_2$. Our method is based on local gluing combined with a global virtual counting argument. The key idea involves a particular design of rod structures to enable a gluing construction and restrict possible degenerations. (2) A topological characterization of algebraically special toric Einstein metrics. This uses special curvature identities for $W^+$ in the toric setting. (3) Uniqueness, up to scaling and isometry, on toric four-manifolds admitting algebraically special Einstein metrics. The proofs uses a variational study of the Einstein–Hilbert functional. (4) Diffeomorphism classification of compact toric Einstein four-manifolds in terms of an improved Hitchin-Thorpe inequality $\chi\geq3|\tau|$. In particular, our construction recovers all the classical toric Einstein metrics, including the Page and Chen–LeBrun–Weber metrics, starting from the round sphere through moduli spaces of conical Einstein metrics.

We consider the inverse problem of recovering a potential $q$, depending on space and time, in the wave equation $(\Box_g+q)u=0$ on a Lorentzian manifold. We prove Lipschitz stability for a formally determined version of this problem under a null-cone foliation assumption. As a consequence, we obtain uniqueness in the Lorentzian Calderón problem for $q$ under this assumption. Our approach is based on a modification of the Bukhgeim–Klibanov method using distorted plane waves.

In the upper half-space model of the hyperbolic space $\mathbb{H}^{n+1}$, a translation hypersurface of type I is a graph $x_{n+1}=f_1(x_1)+\cdots+f_n(x_n)$ and a translation hypersurface of type II is a graph $x_n=f_1(x_1)+\cdots+f_{n-1}(x_{n-1})+g(x_{n+1})$. We classify the minimal translation hypersurfaces of both types for every $n\geq 2$. We obtain these results as particular cases of a classification of the singular minimal translation hypersurfaces of $\mathbb{R}^{n+1}$ for an arbitrary exponent $\alpha\neq 0$, with respect to a direction orthogonal (type I) or parallel (type II) to the hyperplane over which the hypersurface is a graph. The minimal translation hypersurfaces of type I are exactly the parabolic cylinders over a one-parameter family of arches, which contradicts a nonexistence theorem in the literature. The minimal translation hypersurfaces of type II are the parabolic cylinders, the vertical totally geodesic hyperplanes and, when $n\geq 3$, the vertical cylinders over the Scherk surface.

We prove estimates for Gromov's Long Neck Foliated Conjecture in the spin cases, which involves leafwise scalar curvature. The same methods yield a quantitative version of a theorem of Su–Wang–Zhang and a Llarull-type theorem for closed foliated manifolds. A refinement, which involves mean curvature and partially generalizes a theorem of Cecchini–Zeidler, is also obtained.

We construct coupled Dirac-harmonic maps from manifolds of the form $S^1\times M$ where $M$ is a spin manifold equipped with a pair of harmonic spinors satisfying a norm condition. This method produces coupled Dirac-harmonic maps from closed manifolds.

We establish a Bochner formula for double forms in terms of the curvature operator of the second kind. As an application, we prove that a complete Riemannian manifold of dimension $n \ge 4$ with harmonic Weyl tensor and $\frac{3(n-1)}{4}$-nonnegative curvature operator of the second kind is either globally conformally equivalent to a space of positive constant curvature or is isometric to a flat manifold. We prove vanishing theorems for the Lichnérowicz Laplacian $\Delta$ on $(p,q)$ double forms. These generalize recent results of Nienhaus-Petersen-Wink \cite{NPW23} and Dai-Fu-Lu-Yang \cite{DF24,DFY24,FL1,FLD}.

Given a manifold $M$ with an asymptotically cylindrical end, we construct the parametrix for the shifted Laplace operator $\Delta_g + 1$ on the blown-up $b$-double space $M_b^2$. We will review the geometry of manifold with a cylindrical end and construct the inverse directly via separation of variables, which motivates the parametrix construction. Next, we review the mapping properties and show how to get elliptic regularity from the mapping properties of the parametrix. Through this explicit example, we hope this paper provides another reference for the theory of $b$-operators introduced by Melrose \cite{Mel93}.

We give an explicit description of the orbits of the split real form $G_{2(2)}$ on the complex flag manifolds of $G_2$ and study their induced invariant CR structures. In particular we study their Levi forms and show that all possible cases in the simple homogeneous setting occur, namely Levi nondegeneracy, $k$-nondegeneracy for $2\leq k\leq 3$, and holomorphic degeneracy. Moreover, following the classical approach to orbits on flag manifolds, we give equations for these homogeneous CR manifolds.

We remove the log-concavity assumption on the weight in the analytic weighted Yau–Tian–Donaldson correspondence established in [arXiv:2406.10939, arXiv:2407.09929, arXiv:2503.22183] and discuss applications to Sasaki and conformally Kähler Einstein–Maxwell geometries.

To quantize the reduction of a coisotropic submanifold of a Poisson manifold, Cattaneo and Felder constructed a homotopy Poisson algebra together with a deformation quantization. Its degree-zero cohomology is the reduced Poisson algebra, while the full homotopy structure retains information about the formal embedding even when the reduced space is singular or trivial. On the integration side, Cattaneo showed that a coisotropic submanifold integrates to a Lagrangian subgroupoid of any symplectic groupoid integrating the ambient Poisson manifold. It is now recognized that both constructions fit into shifted symplectic geometry: the Lagrangian subgroupoid presents a $1$-shifted Lagrangian morphism of differentiable stacks, and the homotopy Poisson algebra describes the associated $0$-shifted Poisson structure. However, such morphisms form only a special class of $1$-shifted Lagrangians. We extend this picture to $1$-shifted Lagrangians and, more generally, $1$-shifted coisotropics in $1$-shifted symplectic differentiable stacks. We establish an integration theorem for their infinitesimal description in terms of twisted Dirac structures and construct a natural Poisson bracket on invariant functions. Under regularity hypotheses, we equip the associated Lie algebroid complex with a flat $P_\infty$-structure and a curved $A_\infty$-quantization, independent of the auxiliary choices up to isomorphism. The induced bracket in degree zero recovers the reduced Poisson bracket, and suitable cohomological vanishing conditions yield a deformation quantization of invariant functions. This extends the Cattaneo–Felder picture to more general reduction procedures, with the same infinitesimal data governing integration and quantization.

We study a Monge normal form of a surface in $\mathbb{R}^4$ based on the principal axes of its curvature ellipse. The curvature-ellipse-based Monge normal form is obtained using only Euclidean motions of the ambient space and local reparameterizations of the surface, and preserves the Euclidean invariants of the curvature ellipse, including its semi-axis lengths up to their ordering. We also give applications of this normal form to the $2$-jet geometry of surfaces in $\mathbb{R}^4$ and to the geometry of their orthogonal projections.

We establish an almost-splitting theorem for spacetimes under timelike sectional curvature bounds in the spirit of the celebrated Riemannian result due to Cheeger and Colding. More concretely, we show that if a globally hyperbolic spacetime with timelike sectional curvature bounded below by a nonpositive constant close to zero contains a long maximizing timelike segment, then causal diamonds along that segment are close in the Lorentzian Gromov–Hausdorff sense of Minguzzi–Suhr to those in a product metric spacetime. Our result can be interpreted as a quantitative analogue of the Lorentzian splitting theorem for spacetimes with nonnegative timelike sectional curvature by Beem–Ehrlich–Markvorsen–Galloway.

We prove the finiteness of the integral of the top power of the Ricci form on a complete, noncompact Kähler manifold with positive bisectional curvature. This settles a special case of a conjecture of Yau and Yang. An immediate corollary is the finiteness of the fundamental group of such manifolds. As in our previous work, the main idea involves the construction and use of plurisubharmonic weight functions with finite Monge-Ampère masses.

A collection of $m \geq 2$ lines through the origin in $\mathbb{R}^2$ is balanced if the sum of the orthogonal projections onto these lines equals $m/2$ times the identity. A line arrangement in $\mathbb{RP}^2$, endowed with the round metric of curvature $1$, is point-balanced if the tangent lines at every vertex form a balanced collection. A line arrangement $\mathcal{A}$ in $\mathbb{RP}^2$ has the Hirzebruch property if it consists of $3k$ lines and every line contains exactly $k+1$ vertices. Using a Kempf–Ness convexity argument, we show that an irreducible arrangement $\mathcal{A}$ has the Hirzebruch property if and only if its projective equivalence class contains a point-balanced representative, unique up to orthogonal transformations. We then combine elementary properties of balanced collections of lines in $\mathbb{R}^2$ with spherical geometry to prove that every irreducible point-balanced arrangement is a reflection arrangement. This gives a new proof of Panov's classification of real Hirzebruch arrangements.

We prove that every complete, connected, noncompact Riemannian three-manifold without boundary with scalar curvature at least $6$ has conjugate radius at most $2\pi/3$. This resolves Shen's conjecture on the strict conjugate-radius bound $\operatorname{conj}(M,g)<\pi$ in dimension three.

We construct Dirac-harmonic maps from closed spin tori whose map components are not harmonic. For a closed Riemannian surface as target, we prove that such solutions exist for every source metric and every spin structure whenever the target metric is nonflat. The maps can be chosen null-homotopic, of rank one everywhere, with nowhere-vanishing tension and unbounded energy. For a strictly negatively curved closed oriented target surface, every homotopy class contains such a sequence when the source has the trivial spin structure. For the other spin structures we give a compatibility condition for the same construction. Adding a harmonic circle component yields coupled immersions into the product of any such surface with a circle. The construction combines a spinor phase that cancels the rotation of a moving target frame with a curvature balance along a closed curve. Minimizing the energy of short loops with prescribed signed curvature area produces the required curves without any symmetry assumption on the target. Their total geodesic curvature determines the spinor's periodicity. We also give a curve criterion in arbitrary target dimension, applications to totally geodesic surfaces and warped products, and a local choice of target metric that works on every closed manifold of dimension at least two.

The necessary and sufficient conditions for the intersection of two real flag manifolds in a complex flag manifold to be discrete are stated including a previous known result. The discrete intersection of real flag manifolds is the orbit of a Weyl group, which is an antipodal set of a complex flag manifold. The triple of two real flag manifolds and a complex flag manifold as an ambient space is constructed from a compact symmetric triad. It is shown that the compact symmetric triad can be taken to be canonical.

We study the stability of optimal transport for the squared Carnot–Carathéodory distance. We prove the sharp $W_1^{1/4}$ stability estimate for optimal transport maps on complete Riemannian and fat sub-Riemannian manifolds, assuming bounded source support and an upper density bound. On John domains with two-sided density bounds, we obtain logarithmic $L^2$ stability of Kantorovich potentials on complete smooth sub-Riemannian manifolds that are equiregular near the source, with arbitrary compact targets. For potentials, we obtain every Hölder exponent $0<\alpha<1/2$ on two-generating structures, and the exponent $1/2$ in the Riemannian and fat settings and on all step-two Carnot groups. Three-atom examples show that the map and potential exponents with respect to $W_1$ cannot exceed $1/4$ and $1/2$, respectively, on any smooth sub-Riemannian manifold of horizontal rank at least two. We also construct smooth transport families whose target variation of order $t$ in $W_1$ produces map variation of order $t^{1/s}$ in an equiregular region of step $s$. The map estimates follow from coercivity of the duality defect, while the potential estimates follow from quantitative concavity of the Kantorovich functional.

We establish the free-boundary version of Schoen-Simon's regularity and compactness theory for stable minimal hypersurfaces in a complete $(n+1)$-dimensional Riemannian manifold-with-boundary, following the intrinsic PDE scheme introduced by the first-named author. The key step is to prove sheeting theorems in the two possible scenarios in which local weak flatness can occur near a point on $\partial M$: flatness with respect to $\partial M$, and flatness with respect to a reference hypersurface-with-boundary meeting $\partial M$ orthogonally. We obtain these theorems via $\varepsilon$-regularity results for geometrically adapted tilt functions that quantify flatness while keeping track of the ambient geometry. These tilt functions satisfy nonlinear elliptic PDEs on the hypersurface which are amenable to an intrinsic analysis via De Giorgi iteration, made possible by the weak Caccioppoli-type inequalities obtained from stability. Both results apply to immersed hypersurfaces with a singular set of locally finite $(n-2)$-dimensional Hausdorff measure. When specialised to embeddings, they yield compactness and regularity: a sequence of stable free-boundary embedded minimal hypersurfaces with uniformly bounded area and singular sets of locally finite $(n-2)$-measure admits a subsequence converging to a free-boundary minimal hypersurface that is smoothly embedded away from a singular set of Hausdorff dimension at most $n-7$. Convergence is smooth and graphical, possibly with multiplicity, away from this singular set. As an application, we extend to arbitrary dimension the free-boundary Almgren-Pitts existence theory developed by the second-named author with X. Zhou and prove that every compact Riemannian manifold-with-boundary of dimension $n+1$ contains a free-boundary minimal hypersurface smoothly embedded outside a singular set of Hausdorff dimension at most $n-7$.

We provide an alternative and independent proof of the analogue of the Gage-Hamilton-Grayson Theorem for embedded free boundary curve shortening flows in a convex planar domain established by Langford-Zhu. Our proof is based on blowup arguments introduced by B. White in his work on mean-convex mean curvature flow.