Papers

Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$, and let $1\leq k\leq n$. We consider the space of smooth $k$-admissible potentials introduced in the study of the Hessian-cscK equation by Guo–Smith–Tong. The complex $k$-Hessian equation identifies this space, modulo constants, with the space of smooth positive densities of fixed total mass. We compute the differential of the Hessian density map and use it to pull back the Fisher–Rao metric. The resulting Riemannian metric is \[ G^{\HC,k}_{\varphi}(u,v) =\int_X \bigl(\Delta_{\varphi}^{(k)}u\bigr) \bigl(\Delta_{\varphi}^{(k)}v\bigr) \,\omega_{\varphi}^{k}\wedge\omega^{n-k}. \] We then show that the square-root Hessian density map \[ \varphi\longmapsto 2\sqrt{\frac{\omega_{\varphi}^{k}\wedge\omega^{n-k}}{\omega^n}} \] is a global Riemannian isometry onto the positive part of the $L^2(X,\omega^n)$-sphere of radius $2\sqrt{V}$, where $V=\int_X\omega^n$. For $k=n$, this recovers the spherical description of the classical Calabi metric. We also derive the curvature, geodesic and diameter formulas, and prove a quantitative local comparison between adjacent Hessian–Fisher distances.

In this work, we propose a curve to be a minimum of length among rectifiable, continuous curves of unit width in three-dimensional Euclidean space. We show that a previously conjectured minimizer of that problem is not of unit width, give the numerical derivation of a candidate family of unit width curves, prove that it admits a strict local minimum within that candidate family and that this minimum is of unit width and length $3.905124162243593\ldots$. The curve consists of eight straight segments, two cone geodesics, and three circular helices. Its parameters are specified by a uniquely isolated solution of the joining and stationarity equations. To establish unit width in every direction, we combine analytic estimates with interval arithmetic on a finite covering of the remaining directions. We conjecture that the curve is globally minimizing. We supplement this work with the programs used to identify the candidates, replicate the computer-assisted parts of our proof, as well as a formalization of our proof in Lean.

We study the $\alpha$-Ricci flow $(g(t),\alpha(t))$, which is a Ricci flow coupled with a harmonic 1-form heat flow, on the 2-torus $M$. We prove global existence for this flow as well as a Type III curvature estimate. The long-time behavior of the $\alpha$-Ricci flow depends on the cohomology class $[\alpha] \in H^1(M,\mathbb{R})$, which is invariant under the flow. If $[\alpha]$ is rational, then as $t\to\infty$, $(M,t^{-1}g(t))$ Gromov–Hausdorff converges to $S^1$ with length $\ell([\alpha])/\sqrt 2$ where $\ell([\alpha])$ is the primitive length of $[\alpha]$. On the universal cover, the rescaled flow $(s^{-1}\tilde g(st),\tilde\alpha(st))$ smoothly converges to an expander $(g^*(t),\alpha^*(t))$ on $\mathbb{R}^2$ as $s\to \infty$ where $\alpha^*(t)$ is nonzero and constant in spacetime and $g^*(t)$ is flat and expanding in the $\alpha^*$ direction. If $[\alpha]$ is irrational, then there is a sequence $s_i\to\infty$ such that $(M,s_i^{-1}g(s_it))$ Gromov–Hausdorff converges to a point. Moreover, on the universal cover, $(s_i^{-1}\tilde g(s_it),\tilde\alpha(s_it))$ smoothly converges to $(g^*(t),\alpha^*(t))$.

Given a complete and non-compact Riemannian manifold of dimension $3\le n\le 7$, we find conditions for the existence of compact exhaustions $\{U_i\}_{i=1}^\infty$ such that all boundaries $\partial U_i$ have the same sign of mean curvature. In particular, we show that in these dimensions, a manifold with one end satisfying a no tentacle condition always has a weakly mean convex, weakly mean concave, or minimal exhaustion, according to the behaviour of the area profile of separating hypersurfaces. This is a version of a result sketched by Gromov, under a condition which is weaker than his thickness at infinity. In more general situations, we study the existence of exhaustions which are weakly mean concave or weakly mean convex apart from an arbitrarily small $L^1$ error. That is, we find sufficient conditions for the existence of exhaustions $\{U_i\}_{i=1}^\infty$ which satisfy $\int_{\partial U_i}H^+\to0$, $\int_{\partial U_i}H^-\to0$, or $\int_{\partial U_i}|H|\to0$, where $H^-$, $H^+$, and $|H|$ are the negative part, positive part, and the absolute value of the mean curvature respectively. As an application of our methods, we study the existence of almost $c$-convex exhaustions, that is, exhaustions whose boundaries satisfy $\int_{\partial U_i}(H-c)^-\to0$ for some $c>0$.

We study reversible Higgs bundles on a compact Riemann surface $\Sigma$ of genus $g\geq 2$, that is, $G$-Higgs bundles $(E,\Phi)$ with $(E,\Phi)\cong(E,-\Phi)$. For a complex reductive group $G$ realized as a self-adjoint subgroup of $GL(n,\mathbb{C})$, we show that if $(E,\Phi)$ corresponds to a representation $\rho$ of $\pi_1(\Sigma)$, then $(E,-\Phi)$ corresponds to $(\overline{\rho}^{\,t})^{-1}$. As a consequence, the moduli space of reversible Higgs bundles is the fixed locus of an involution of the character variety and carries the structure of a real affine algebraic set. For a connected semisimple group and every topological type, we prove that the reversible locus in the smooth moduli space is a $(B,A,A)$-brane. We then describe reversibility through spectral data: in even rank, for a smooth spectral curve, reversible Higgs bundles correspond to line bundles invariant under the involution $\eta\mapsto-\eta$; in odd rank, under natural genericity assumptions for which the spectral curve is nodal, we give a gluing criterion together with an explicit polystability condition. Finally, for $SL(2n,\mathbb{C})$ we construct a relative Fourier–Mukai transform over the regular locus of the Hitchin base and prove that it maps the structure sheaf of the reversible locus to an object whose support is exactly the regular degree-zero $PSp(2n,\mathbb{C})$-locus in the Langlands dual $PGL(2n,\mathbb{C})$-moduli space. This support is of type $(B,B,B)$.

Motivated by the standard 2-dimensional 3-segment snake model, leading to the famous $(2,3,5)$ geometry, we discuss a series of 3-dimensional models of similar snakes. In particular, we seek for rules allowing to steer them exclusively by the two joints, which expects 4-dimensional control freedom. This leads to interesting $(4,9)$ and $(4,10)$ models. Finally, we discuss modifications of the previous snakes which lead to $(4,8)$ models. All of the models open further questions and should be a good baseline for further research.

Let $X:M^3\to\R^4$ be a complete, connected self-shrinker without boundary, and suppose that the squared norm $S$ of its second fundamental form is constant. We prove that $S=0$ or $S=1$. Accordingly, the image is a hyperplane through the origin or a generalized cylinder $\mathbb S^k(\sqrt{k})\times\R^{3-k}$, $1\le k\le3$, and the immersion is a Riemannian covering onto its image.

An exterior free-boundary minimal surface is a properly immersed minimal surface in the complement of the open unit ball of $\mathbb{R}^3$ whose boundary lies on the unit sphere and meets it orthogonally. We prove that for all integers $\ell\geq2$, $b\geq1$ and $\gamma\geq0$ there is a connected, orientable, properly embedded exterior free-boundary minimal surface with $\ell$ ends, $b$ boundary components and genus $\gamma$. Its total curvature is finite, and its ends are parallel catenoidal ends of multiplicity one. By the theorem of Mazet and Mendes, a surface of this kind with one end is a rotational annulus. Hence the topological types of such embedded surfaces are exactly the annulus with one end and all types with at least two ends. There are two gluing constructions. For $\ell\geq3$ we stack $\ell$ sheets over a stable catenoidal annulus that meets the sphere along a circle of radius $4/5$, and we join adjacent sheets by half-catenoidal bridges sitting on the sphere. The sheets are small normal graphs over this annulus whose displacements contain different multiples of the Jacobi field of the family of such annuli, so their boundary circles and the logarithmic growth of their ends differ slightly. For $\ell=2$ we double a catenoidal annulus near the critical angle $\pi/4$, where it acquires a bounded rotationally invariant Jacobi field; the distance to the critical annulus is coupled to the size of the bridges. In gluing constructions inside the unit ball that double or stack a compact surface, the bridges themselves must balance the sheets, which forces many bridges. Here the logarithmic growth of the ends absorbs this balancing. A fixed number of bridges therefore suffices, and every topological type is reached, including the small ones.

Let $\Gamma<SO(4)$ be a nontrivial finite group acting freely on $S^3$, and let $M=S^3/\Gamma$. We prove that every smooth conformal class $C$ on $M$ satisfies \[ Y(M,C)\le |\Gamma|^{-2/3}Y(S^3), \] with equality for the quotient-round conformal class. Consequently, \[ \sigma(M)=|\Gamma|^{-2/3}Y(S^3). \] Our approach uses four-dimensional Poincaré–Einstein fillings of the ball. We first establish compactness for fillings whose conformal infinities range over a compact family of positive-scalar-curvature metrics. Since compactness alone does not guarantee continuation through degenerate Einstein metrics, we use equivariant Fredholm parity to obtain a filling with the prescribed boundary conformal class and group action. A common interior fixed point then makes the Li–Qing–Shi boundary test function invariant, and its descent to $M$ gives the sharp covering factor.

We study the invariant rank-four deformed Hermitian–Yang–Mills equation associated with the chain $\mathcal{O}(1)\to\mathcal{O}(2)^{\oplus2}\to\mathcal{O}(3)$ over $\mathbb P^1$, with vertical weights $(4,2,0)$ and a fixed product Kähler class and phase. For arbitrary linear arrows $\psi=Bz$ and $\phi=(Az)^T$, smooth strictly calibrated reduced solutions exist exactly when $A$ is invertible and $z^TA^TBz$ is not identically zero. The same condition characterizes stability, semistability and polystability in the specified equivariant coherent category. The classification includes all eleven nonempty coefficient strata and both coincident and distinct divisor points on the rank-one lower boundary. The analytic proof retains the ordered, noncommuting middle equation. After constructing positive-parameter solutions, it rules out unbounded metric values by combining the source budgets with moving spectral tests; on the rank-one lower boundary, the divisor mass must also be retained. Global value, gradient and symbol estimates then give a smooth strictly calibrated endpoint on the original chain. No equivalence with full coherent or derived-category stability, uniqueness, or estimate uniform across changing chains is asserted.

We associate each h-convex body in hyperbolic space $\mathbb{H}^{n+1}(n\geq 2)$ with a Radon measure valued $(0,2)$-tensor on the sphere $\mathbb{S}^n$ through its horospherical function. Using spherical convolution, we prove that any h-convex body can be approximated by a sequence of smooth strictly h-convex bodies in the Hausdorff sense. This approximation result extends the inequality parts of the Alexandrov-Fenchel inequality and the modified Alexandrov-Fenchel inequality to general h-convex bodies, but equality rigidity cannot be obtained directly. We prove that the equality holds in the Alexandrov-Fenchel inequality and the modified Alexandrov-Fenchel inequality if and only if the h-convex bodies are geodesic balls. For the Alexandrov-Fenchel inequality, we establish the Hausdorff stability independent of curvature bounds described by the Alexandrov-Fenchel deficits for h-convex bodies, which implies the equality rigidity of the Alexandrov-Fenchel inequality. For the modified Alexandrov-Fenchel inequality, we introduce a class of globally constrained $k$-th modified quermassintegrals preserving flows with general h-convex initial hypersurfaces to obtain the equality rigidity. Moreover, we prove the flows above preserve h-convexity, make the $(k+1)$-th modified quermassintegrals non-increasing and converge to geodesic spheres. We prove a rigidity theorem for modified area measures for h-convex bodies and use this rigidity theorem to characterize the geodesic balls via modified support measures for h-convex bodies, from which we prove that the $(k+1)$-th modified quermassintegrals strictly decrease in a short time along the flows with non-spherical initial hypersurfaces. Then we obtain the equality rigidity of the modified Alexandrov-Fenchel inequality.

In general, the capillarity energy outside an unbounded obstacle may fail to admit volume-constrained minimizers, as infimizing sequences may escape to infinity. We study this minimization problem outside a closed convex obstacle $C$. Via a $\Gamma$-convergence result for the capillarity energies outside sequences of closed convex obstacles converging in the Hausdorff sense, we prove that the infimum of the functional is attained by a generalized minimizer, namely, a finite union of regions, each of which minimizes the capillarity energy outside a limit of translations of $C$ in its volume class, with the total volume satisfying the prescribed constraint. Each generalized minimizer can be suitably approximated by an infimizing sequence, thus providing a description of its asymptotic behaviour. As an application, we study the isoperimetric profile associated with the capillarity energy and investigate how the asymptotic geometry of the obstacle affects both this profile and generalized minimizers and their connectedness.

Let $n\ge5$ be odd and let \(L_{3q}^n=S^n/C_{3q}\) be the standard lens space, where $q>3$ is prime. We prove that for all sufficiently large $q$, \[ \sigma(L_{3q}^n) > (3q)^{-2/n}Y(S^n)=Y(L_{3q}^n,[g_{\rm rd}]). \] Here \([g_{\rm rd}]\) denotes the quotient-round conformal class. Combining spin bordism with the surgery theorem of Ammann–Dahl–Humbert, we establish \[ \sigma(L_{3q}^n)\ge \min \{q^{-2/n}Y(S^n),\Lambda_n\}, \] where $\Lambda_n>0$ depends only on $n$. For sufficiently large $q$, this exceeds the quotient-round value, showing that the quotient-round conformal class need not maximize the smooth Yamabe invariant in any odd dimension $n\ge5$.

We extend Uhlenbeck's classical results on the generic properties of Laplacian eigenfunctions. In particular, we find a residual set of $C^k$ domains in $\mathbb{R}^n$, $k > n$, such that the nodal set of any Dirichlet eigenfunction intersects the boundary orthogonally along a $C^{k-1}$ embedded submanifold, whenever the intersection is non-empty. This is achieved by analyzing the boundary asymptotics of Dirichlet eigenfunctions via homogeneous harmonic polynomials and comparing their respective nodal sets. Along the way, we give a complete proof that generic Euclidean domains have simple Dirichlet eigenfunctions with zero as a regular value in the interior and zero as a regular value of the boundary normal derivative. In the special case of planar domains, a different argument provides one-parameter families of perturbed domains with the desired properties.

We establish a Weyl law for the codimension-one anisotropic volume spectrum on compact Riemannian manifolds. We then use this law to obtain equidistribution in the full Grassmannian bundle while keeping the ambient metric fixed. More precisely, on every closed connected smooth Riemannian manifold of dimension three through six, we construct smooth, embedded hypersurfaces stationary for anisotropies smoothly converging to area. The cumulative sum of their area measures, when lifted to the Grassmannian bundle, converges to the uniform measure.

Let $F:M^n\to\mathbb{S}^{2n+1}$ be a minimal Legendrian immersion of a connected closed manifold, where $n\ge3$, and let $B$ be its second fundamental form. We prove that if $|B|^2\le(n-1)(n+2)/n$, then, up to a unitary transformation, $F$ parametrizes a totally geodesic sphere or is a finite covering of the image of the minimal Calabi product of a point and a totally geodesic Legendrian sphere. This proves Conjecture 1 of Luo and Sun [Ann. Inst. H. Poincaré C Anal. Non Linéaire 40 (2023), p. 548].

We consider the image of the comparison homomorphism from bounded cohomology to ordinary cohomology. Its dimension is called the bounded Betti number. In our earlier work we proved that the total bounded Betti number of a closed Riemannian $n$-manifold satisfying $\Ric\geq -(n-1)$ and $\Diam\leq D$ is bounded by a constant depending only on $n$ and $D$. The original proof used the amenability of the images of sufficiently small local fundamental groups. Since that work was written, Kapovitch and Wilking proved Gromov's generalized Margulis lemma under a lower Ricci curvature bound. Their theorem gives the stronger conclusion that these local groups are uniformly virtually nilpotent. We explain how their result supplies a direct modern input for the bounded Betti number theorem. We also discuss the topological meaning of the bounded part of ordinary cohomology and its relation to the $\ell^1$-seminorm on homology.

We construct non-holomorphic Siegel modular forms whose coefficients encode linking numbers in arithmetic hyperbolic manifolds of simplest type. Our proofs rely on an improved understanding of the transgression forms appearing in Kudla–Millson theory, and along the way we establish some structural properties including in particular their expected current equations.

We construct smooth functions on the flat torus $\mathbb{T}^{n\ge5}$ that change sign and have negative mean but cannot be realized as the scalar curvature of any metric in the given conformal class.

In this paper, we introduce a concept of modified curvature measures for h-convex domains in hyperbolic space and prove a shifted version of the generalized Alexandrov theorem, which characterizes geodesic balls in hyperbolic space via modified curvature measures. This result generalizes the classical work of Kohlmann (1998). We further investigate a volume-preserving shifted $\alpha$-Gauss curvature flow for strictly h-convex hypersurfaces in hyperbolic space and prove that for any $\alpha>0$ the solution exists for all time and converges smoothly to a geodesic sphere. The theory of modified curvature measures developed in the first part serves as a key ingredient in characterizing the Hausdorff limit shape of the flow.

We prove an analogue of Min-Oo's conjecture for sufficiently small spherical caps in the unit round sphere. For every dimension $n\ge3$, there exists $r_*(n)>0$ such that the following holds for every round cap of radius $0<r\le r_*(n)$: any smooth compact connected oriented Riemannian fill-in with the same intrinsic boundary metric and with scalar curvature and outward boundary mean curvature at least those of the cap is isometric to the cap.

In this paper, we prove a uniform pseudolocality theorem for smooth complete Ricci flows whose initial metrics have Ricci curvature bounded below and uniformly positive injectivity radius. No a priori curvature bound is assumed on the flow. As a consequence, we obtain strong uniqueness for smooth complete Ricci flows from a complete metric with the above geometric bounds. We also prove that every smooth complete Ricci flow from a complete Ricci-flat metric with uniformly positive injectivity radius is stationary. The same conclusion holds for any complete flat initial metric. In particular, we give an affirmative answer to Chen's question on the strong uniqueness of Ricci flow from Euclidean space in every dimension.

Let $M^n$ be a closed simply connected Legendrian submanifold of $\mathbb S^{2n+1}$, where $n\geq5$, and let $\mathbf{B}$ and $\mathbf{H}=\operatorname{tr}\mathbf{B}$ denote its second fundamental form and mean-curvature vector. We prove that $M$ is homeomorphic to $\mathbb S^n$ if \begin{equation*} |\mathbf{B}|^2\leq \frac{(n-1)(n+2)}n+\frac{n^2+3n-2}{2n^2}|\mathbf{H}|^2-\frac{(n-1)(n-2)}{2n^2}|\mathbf{H}|\sqrt{|\mathbf{H}|^2+4n}. \end{equation*} The equality can be attained by the Calabi products, so simple connectivity cannot be omitted. For $n\geq4$, we also prove a sphere theorem under an explicit bound and the weaker assumption $b_1(M;\mathbb R)=0$. The proof combines phase-rotated Lawson–Simons variations with a systolic inequality for finite two-complexes. We also give local constructions showing that the pinching condition allows negative isotropic curvature at a point.

We construct $T^2$-invariant Einstein metrics with positive Ricci curvature on #_m(S^2 x S^2) for every integer $m>1$. These provide infinite family of Einstein metrics on closed $4$-manifolds with $\lambda>0$, which are in addition not Hermitian. We further establish the uniqueness of the $T^2$-invariant Einstein metrics on $S^4$, CP^2, S^2 x S^2, and CP^2#\bar{CP}^2.

We prove that every tangent flow at the first singular time of a smooth, closed, embedded mean curvature flow has smooth support, provided that the total curvature satisfies $\sup_{t\in[0,T)}\int_{M_t}|A|^n\,d\mu_t<\infty$. This extends Ilmanen's smoothness result for tangent flows to higher dimensions.

It is observed that the dimension of the space of invariant spinors on the manifold $\mathrm{SU}(n)/T^{n-1}$ of full flags in $\mathbb{C}^n$ equals the number $\mathrm{RT}(n)$ of labelled regular tournaments of order $n$, for $n$ odd. This space is decomposed under a double cover of the Weyl group, and realized as a pair of eigenspaces, first for the Dirac operator, then for a twisted version of it which acts equivariantly; this is used to show that multiplicities of irreducible representations are even for $n\equiv 1\pmod 4$.

In this paper, we first construct infinitesimal equivariant analytic torsion forms for Kähler fibrations. We then compare them with Ma's equivariant analytic torsion forms, modulo $\partial$- and $\overline{\partial}$-exact forms, in terms of equivariant Bott–Chern currents and the $I$-genus. Our comparison formula extends that of Bismut and Goette to families.

We construct a stationary weakly harmonic map from a domain in \R^5 into a compact Kähler surface for which a tangent map is weakly harmonic but nonstationary. The target manifold is the blow-up of \CP^1\times\CP^1 at four points lying in one fiber and arranged at the vertices of a regular tetrahedron. The nonstationarity of the tangent map arises from the break down of the tetrahedral symmetry in the defect measure concentrated on the vertices.

We show that every symplectic manifold admits a symplectic form whose metaplectic-c prequantizations give quantizability conditions on all its Lagrangian submanifolds. Metaplectic-c prequantizations descend to prequantizations over the Lagrangian Grassmannian bundle that are fibrewise flat with holonomies primitive fourth roots of unity on the generators of the fundamental groups. This fibrewise holonomy is consistently interpreted as that of both Leray two-orientations and normalized Blattner–Kostant–Sternberg half-forms. The prequantizations pull back to flat circle bundles on each Lagrangian submanifold whose holonomy character defines a quantizability condition. The first and second Stiefel–Whitney classes are identified as splitting obstructions to the Lagrangianwise holonomies. These holonomies split into a product of holonomies on any orientable Lagrangian submanifold, and split consistently for all Lagrangian submanifolds of a manifold with metaplectic structure. When the first Chern class vanishes, the holonomy splits into the Bohr–Sommerfeld factor and a quarter phase Maslov correction, recovering the classical Einstein–Brillouin–Keller condition. We develop the underlying obstruction theory for circle extensions associated to general Lie group double coverings with first hypercohomology groupoids.

In this paper, we prove that every Hermitian conformal class on a connected noncompact complex manifold of dimension at least two contains a Gauduchon metric. We give a sufficient condition for uniqueness of such Gauduchon metrics up to scaling in terms of functions bounded from above and their Levi forms. This condition is necessary when the manifold has at least two ends. In that case, uniqueness in a fixed conformal class is equivalent to the assertion that every positive solution of the complex Laplace equation for a metric in that class is constant. For a connected noncompact complex Lie group of dimension $n$, uniqueness in every conformal class is equivalent to the assertion that, for every Hermitian metric, every positive solution of the complex Laplace equation is constant. These properties hold exactly when the complex tangent space of a maximal compact subgroup has dimension $n-1$; such a group is abelian. In a conformal class containing a balanced metric, the same equivalence holds without a restriction on the number of ends. We also prove uniqueness for manifolds with a suitable potential on a compactification with compact fibers over a point, the real line, or a parabolic Riemann surface. The examples include compact pluripolar deletions, partially deleted compact fibers, and semipositive line bundles. We construct conformal Gauduchon metrics which are not related by scaling on Stein manifolds and on the total space of $\mathcal O_{\PP^1}(1)^{\oplus2}$.