Papers

We show that the systolic area of every nonpositively curved closed surface $M$ other than the torus is at least $1$, with equality if and only if $M$ is isometric to a square flat Klein bottle. The proof focuses on the Klein-double $4\mathbb RP^2$ and exploits Weil's isoperimetric inequality and a comparison theorem involving a new kind of exponential-type map.

Every real vector bundle of rank at least seven over $CP^3$, or every real vector bundle of rank at least eight over $CP^3\#CP^3$, admits a complete metric of nonnegative sectional curvature on its total space. The corresponding unit sphere bundles also admit nonnegative sectional curvature. These are fixed-rank results: no additional trivial summand is required, and there is no restriction on any characteristic class. The proof combines low-dimensional bundle classification with the Grove–Ziller lifting theory over $S^4$ and $CP^2$.

We study transversely projectively flat regular foliations on compact Kähler manifolds.

A complete one-ended three-manifold with strictly positive Ricci curvature is constructed such that $$ \limsup_{R\to\infty}\frac1R\int_{B(p,R)} Scal\,dV=+\infty. $$ The construction combines an explicit toric lens carrying a large scalar-curvature integral with a three-dimensional angular pair of pants and an adaptive sequence of hybrid blocks.

Let $(M^n,g_0)$ be a closed smooth Riemannian manifold, with $n\geqslant 3$. We prove short–time existence and uniqueness for the "critical" Ricci–Bourguignon flow \begin{equation} \partial_t g=-2\operatorname{Ric}_g+\frac{R_g}{n-1}g\,. \end{equation} Up to a constant rescaling of time, this is the Schouten flow, since the right–hand side of the equation is $-2(n-2)$ times the Schouten tensor. At the critical parameter, however, the principal symbol of the linearization of the operator that we obtain after the usual "DeTurck modification" still has a zero eigenvalue. We resolve this degeneracy by adjoining an independent scalar unknown $r$, intended to represent the scalar curvature, and considering an "extended" system consisting of a strictly parabolic equation for the metric coupled to a transport–reaction equation for $r$ with a curvature–square forcing. The extended system belongs to the class of nonlinear composite parabolic–hyperbolic systems studied by Vol'pert and Khudyaev~\cite{VolpertKhudyaev1972}. Then, a standard compact–manifold adaptation of their theorem yields a unique smooth solution of the extended system. Finally, the difference $R_g-r$ satisfies a homogeneous linear parabolic equation, so the scalar–curvature constraint "propagates" and the original critical Ricci–Bourguignon flow is recovered. This settles the short–time existence and uniqueness problem at the critical “Schouten” value $\rho=1/(2(n-1))$, which was left open by the previous Ricci–Bourguignon theory in~\cite{CatinoEtAl2017}.

We use the Kähler-Ricci flow to give alternate proofs of several known Miyaoka-Yau inequalities and slope semi-stabilities, in particular, for the case of compact Kähler manifolds with semi-ample canonical line bundles and K-semistable Fano manifolds. Moreover, when the canonical line bundle is semi-ample, and the Kodaira dimension is $n-1$, we prove the Miyaoka-Yau quantity is equal to the intersection of the Weil-Petersson metric and the Fubini-Study metric on the canonical model. Consequently, equality holds in the Miyaoka-Yau inequality if and only if the pluricanonical map is a holomorphic fibre bundle.

In the case of the complex hyperbolic space, we establish a version of the hyperbolic Ax–Lindemann Theorem for the topological closure rather than the Zariski closure. Our approach combines classical differential geometry with homogeneous dynamics.

We complete the classification of compact simply connected homogeneous Einstein manifolds in dimension six by resolving the remaining cases for left-invariant Einstein metrics on $G=\mathrm{SU}(2)\times\mathrm{SU}(2)\cong S^3\times S^3$. Previous work leaves two cases for the isotropy group $K$, namely $K=\{e\}$ and $K\cong\mathbb Z_2$. We first rule out $K=\{e\}$. Then we show that any $\mathbb Z_2$ symmetry generated by an inner involution $\sigma$ with $\tr\sigma=-2$ necessarily extends to a $\mathbb Z_2\times\mathbb Z_2$ symmetry. Together with the previously known classification results, these theorems imply that every left-invariant Einstein metric on $G$ is, up to homothety and isometry, either the standard product metric $g_{\rm can}$ or the Jensen nearly Kähler metric $g_{\rm NK}$.

Let $M^5$ be an oriented real hypersurface of a six-dimensional strict nearly Kähler manifold $(\widetilde M^6,\widetilde g,J)$, and let $\xi=-JN$ and $\eta=g(\,\cdot\,,\xi)$ denote the induced Reeb field and Reeb one-form. We prove the pointwise obstruction \[ (d\eta)_p\neq0\qquad\text{for every }p\in M. \] The proof is local and uses only the hypersurface identity \[ \nabla_X\xi=\phi AX-G(X,N) \] together with the six-dimensional strict nearly Kähler algebra. If $(d\eta)_p=0$, then \[ \phi A+A\phi=2G_N,\qquad G_NX=G(X,N), \] which forces the Hopf condition. On the contact hyperplane $\mathcal D_p=\ker\eta_p$, the endomorphism $\phi A+A\phi$ then commutes with $\phi$, whereas $G_N$ anti-commutes with $\phi$ and satisfies \[ G_N^2=-\alpha I_{\mathcal D_p},\qquad \alpha>0. \] This contradiction proves the obstruction. As a connection-theoretic consequence, for the standard Yano semi-symmetric metric connection associated with $\xi$, \[ \widehat\nabla_XY=\nabla_XY+\eta(Y)X-g(X,Y)\xi, \] one has in dimension five \[ \operatorname{Skew}\operatorname{Ric}^{\widehat\nabla}=-\frac32\,d\eta, \] and hence $\operatorname{Ric}^{\widehat\nabla}$ is nowhere symmetric. The obstruction also rules out any local induced almost contact class whose defining conditions include $d\eta=0$. We illustrate the result on the standard homogeneous strict nearly Kähler spaces.

We study gradient Ricci solitons on partially totally real (PTR) submanifolds of complex space forms. In nonzero ambient holomorphic sectional curvature, the Gauss–soliton equation reconstructs the square $P^2$ of the tangential Kähler operator from the Hessian endomorphism and an extrinsic Ricci operator. Consequently, the PTR tangency spectrum is encoded by intrinsic soliton data together with the second fundamental form. For compact minimal PTR solitons with positive ambient curvature we obtain $\lambda\le c(n-1)/4$, with a strict version under a relative-nullity condition. We also prove compact triviality criteria, an exact trace-free anisotropy identity, a parallel-data rigidity result, and a nonexistence theorem for compact real hypersurfaces. A flat multi-level Gaussian model and a nonflat Segre $CR$-product provide explicit consistency checks and identify the limiting cases of the reconstruction formulas.

We show that on a compact, connected, Riemannian manifold without boundary, every isotopic to the identity diffeomorphism can be written as a finite composition of optimal mass transport maps. This result represents a non-linear infinite dimensional generalization of Ballantine's factorization results.

Let $M$ be an $n$-dimensional complete connected self-shrinker in $\mathbb{R}^{n+p}$. Denote by $\operatorname{Ric}$ and $\mathbf{H}$ the Ricci curvature tensor and the mean curvature vector of $M$, respectively. We prove that if the self-shrinker $M$ satisfies $\operatorname{Ric}\ge(\frac{n-2}{n^2}+\varepsilon_n)|\mathbf{H}|^2g$, then $M$ is either a linear subspace or a round shrinking sphere, where $\varepsilon_n$ is an explicit positive constant equal to $\frac{1}{3n^2}+O\left(\frac{1}{n^5}\right)$. Moreover, we obtain a rigidity theorem that is sharp in codimension two for the self-shrinker satisfying $\operatorname{Ric}\ge\frac{n-2}{n^2}|\mathbf{H}|^2g$.

We construct an endpoint map with no regular values, disproving the sub-Riemannian Sard conjecture. Remarkably, in our construction the minimizing Sard property remains valid.

We establish a Cheng-Yau type estimate for positive biharmonic functions on complete Riemannian manifolds with Ricci curvature satisfies $Ric_g \ge -(n-1)K g$. If $u$ is a positive biharmonic function in $B_{2R}(p)$, then $$ -\frac{\Delta_g u}{u} +\frac{1}{8n}\frac{|\nabla u|_g^2}{u^2} \le C_n\left(R^{-2}+K\right) \quad\text{on }B_R(p). $$ Further, if the Ricci curvature is nonnegative, every global positive biharmonic function satisfies $\Delta_g u\equiv c$ and the sharp estimate $|\nabla u|_g^2\le2cu$ for some nonnegative constant $c$. We also show that every positive $k$-polyharmonic function has nonnegative constant $(k-1)$-st Laplacian and growth of order at most $2k-2$.

We prove that no product of positive-dimensional compact complex manifolds admits a strict locally conformally Kähler (lcK) metric. This solves a long-standing conjecture in lcK geometry, previously established only under certain conditions.

We construct the Cartan calculus of differential cubical forms on an object $X$ of a cartesian tangent category with a scalar multiplication by a commutative ring object $R$. We prove that the differential cubical forms are naturally equipped with the structure of a commutative differential graded algebra. Then we show that, if $R$ has no $2$-torsion, this CDGA together with the Lie algebra of vector fields and the inner derivatives constitutes a Cartan calculus, which lies between the initial Cartan calculus of algebraic Kaehler forms and the terminal one of Lie-Rinehart forms. We give a number of examples, including affine schemes and elastic diffeological spaces, and show that we recover the established notions of differential forms in all cases.

We prove that every smooth vector field on a closed smooth manifold is a single Lie bracket. The proof combines a relative transport construction on the complement of a ball with an exact extension across transverse graph disks. For the local extension, we first arrange that an auxiliary bracket is transverse to the disks, straighten it by its flow, and then remove the resulting scalar discrepancy. The argument applies without an orientability assumption. It also implies that every smooth fiberwise linear function on the cotangent bundle is a Poisson bracket of two such functions.

In this paper, we consider compact locally conformal Kähler manifolds with constant $k$th-mixed curvature. By using a recent method of Huang-Wan for constant Chern holomorphic sectional curvature, we prove that if a compact LCK manifold has nonzero constant $k$th-mixed curvature, then its Hermitian metric is Kähler. For the second mixed curvature, the same conclusion also holds when the curvature constant is zero. We also study some special parameters related to the general constant mixed curvature conjecture. In particular, we obtain a torsion-energy identity and a sign obstruction for compact Hermitian manifolds, and characterize an exceptional Kähler case by Bochner–Kähler geometry.

Let $(M^m,g)$, $m\geq2$, be a closed connected Riemannian manifold with $\secg_g\leq-1$, and let $(X,g_X)$ be its universal cover. Yau proved that $\hiso(X)\geq m-1$. We prove that equality is rigid: if $\hiso(X)=m-1$, then $(X,g_X)$ is isometric to $\Hh^m(-1)$.

The flatness of the under-actuated unmanned surface vessel with circular hull-shape (“the hovercraft") has long been known. We resolve the long-standing open problem: whether the generic under-actuated 6-degrees-of-freedom (dof) unmanned surface vessel (gUSV) admits a feedback linearization. Lie symmetries of the gUSV are used to construct a dynamic feedback linearization, explicit flat outputs and flat parametrization, for all hydrodynamic parameter values. In this generic case the dynamic compensator has order 6, in contrast to the hovercraft vessel whose dynamic compensator has order 4.

On \[ S^5\times S^2,\qquad S^5\times S^3,\qquad S^5\times S^4,\qquad S^7\times S^2, \] we construct infinitely many pairwise non-isometric real-analytic Einstein metrics. After discarding finitely many members, none is isometric or homothetic to a Riemannian product or to a classical two-summand Böhm metric. The metrics have the form \[ dt^2+a(t)^2g_{S^p}+b(t)^2g_{S^p}+c(t)^2g_{S^q}, \qquad p,q\ge2,\qquad 2p+q\le8. \] The construction starts with solutions for which the two non-collapsing radii at one singular orbit are very small. After magnifying this region, the Einstein equations approach a Ricci-flat problem whose solutions tend to a cone by a decaying oscillation. The main point is to determine exactly which part of this oscillation can still be seen by the closing conditions at the other end of the manifold. Rather than following the full, increasingly oscillatory shooting map, we extract two response quantities from the Ricci-flat limit. One determines the common oscillatory phase; after that phase has been removed, the second closing condition is reduced to a scalar sign change. A uniform inner–outer matching argument transfers this limiting sign change to the exact Einstein equations. As the initial scale tends to zero, the phase makes arbitrarily many turns, and the closing conditions are therefore met infinitely many times.

We describe the topology of the set of tridiagonal symmetric real matrices with fixed non-simple spectrum. The tridiagonal isospectral set is not a submanifold of the corresponding orthogonal conjugacy class, but a collection of submanifolds which intersect cleanly. We show that the tridiagonal isospectral set is an aspherical space and that its Betti numbers are combinatorial numbers. This generalizes to arbitrary spectrum constructions and results known for simple spectrum. We also extend these results to split real semisimple Lie algebras.

Let $(M,g)$ be a closed Riemannian manifold of positive Ricci curvature. Let $f$ be a proper isoparametric function on $M$, and $\Gamma$ the unique level set of $f$ which is a minimal hypersurface. For any positive integer $k$, and $\lambda >0$ large enough, we construct a solution of the Allen-Cahn equation $ \Delta u + \lambda (u-u^3 ) =0$ which is constant along the level sets of $f$ and has exactly $k$ nodal components. We prove that the nodal components approach the minimal isoparametric hypersurface $\Gamma$ as $\lambda \rightarrow \infty$ and the energy is uniformly bounded for all such solutions.

We introduce, motivated by noncommutative Carrollian geometry, the notion of a Carrollian–Poisson bundle understood as a Carrollian manifold with a compatible Poisson structure. Compatibility is expressed as the inclusion of the Carroll distribution in the image of the Poisson anchor. We explore the geometric consequences of this compatibility and present several simple examples of Carrollian–Poisson bundles, including the (outer) Kerr horizon, the BTZ horizon, and future null infinity. Furthermore, we establish that every Carrollian–Poisson bundle can be equipped with a Vaisman (contravariant) connection that is both metric-compatible and clock-compatible, and thus geometric kinematics can be defined and studied.

We complete the local classification, up to isometry, of $2$-dimensional pseudo-Riemannian metrics (i.e. both Riemannian and Lorentzian), admitting a projective symmetry algebra of dimension at least two. The new contribution is the treatment of the non-regular case: we obtain a complete list of mutually non-isometric normal forms in neighbourhoods of points where the action of the projective Lie symmetry algebra fails to be regular, including all singular behaviours that can occur. Together with the known classification in the regular case, this completes the solution of the problem posed by Sophus Lie in 1882.

Quandles can be regarded as generalizations of symmetric spaces. In the theory of symmetric spaces developed by Chen and Nagano, there is an interesting relationship between the two-number and the Euler characteristic. The two-number is a Riemannian geometric invariant that can also be characterized in terms of point symmetries, whereas the Euler characteristic is a topological invariant. The aim of this paper is to initiate the study of quandle analogues of Chen–Nagano theory. In particular, we investigate relationships between the two-numbers and the Euler characteristics of quandles, and provide examples of finite quandles that either satisfy or fail to satisfy properties analogous to those of symmetric spaces. These examples are constructed from directed simple graphs labeled by abelian groups.

Motivated by the strong cosmic censorship conjecture in general relativity, we prove the inextendibility of spherically symmetric weak null singularities as spherically symmetric Lorentzian manifolds with a continuous metric and Christoffel symbols in $L^{s}_{\text{loc}}$ for $s>1$. The result assumes a suitable blow-up condition on the derivative of the area-radius function transverse to the weak null singularity in combination with a rigidity result on continuous spherically symmetric extensions across null boundaries proven in [1]. In particular we show that these assumptions are satisfied by the Reissner-Nordström-Vaidya spacetime as well as by a class of spacetimes arising from small and generic spherically symmetric perturbations of subextremal Reissner-Nordström under the Einstein-Maxwell-scalar field system.

In this paper, we prove that if $E$ is a pseudo-effective sheaf with vanishing first Chern class on a klt compact Kähler space $X$, then, after passing to a finite quasi-étale cover, the reflexive pullback of $E$ is locally free and flat. This extends the flatness criterion of Höring–Peternell, originally established for projective varieties, to the Kähler setting. The proof relies on two main ingredients, both of which are new even in the projective case. The first is a flatness theorem for stable sheaves: we show that a slope-stable pseudo-effective sheaf with vanishing first Chern class is Hermitian flat. This is obtained by combining Hermitian–Einstein theory with the subharmonicity properties of direct image sheaves. The second is a singular Kähler analogue of Simpson's flatness theorem for extensions of locally free Hermitian flat sheaves.

Inspired by the monotonicity formula of Alt, Caffarelli, and Friedman, we consider a natural variant of the ACF functional in positively curved 2-dimensional settings where we integrate over sublevel sets of Green's function rather than disks. We prove sharp almost-monotonicity formulas for our new functional in the case of convex planar domains (both with the pole on the boundary and in the interior) and in the setting of 2-dimensional complete manifolds with Euclidean volume growth and nonnegative Gaussian curvature. The tools include the Schwarz-Christoffel formula and Riesz decomposition using conformal coordinates. As a consequence, using quasiconformal estimates, we give a new Lipschitz bound in the manifold setting for minimizers of the two-phase free boundary problem of Alt, Caffarelli, and Friedman, with the constant depending on only the total integral curvature, in contrast to work of Teixeira and Zhang, which gives constants depending on pointwise bounds for the Riemann curvature tensor and its derivatives. Our methods also recover the Lipschitz bound up to a Neumann boundary of Gemmer, Moon, and Raynor in the planar convex domain case.

In this paper, we study the Calabi curvature operator on Kähler manifolds. First, we prove that if the Calabi curvature operator on Kähler manifolds satisfies $\frac{{n\left(n + 1\right)}}{2}$-positive (nonnegative), $\frac{{n + 1}}{2}$-positive (nonnegative), and $\left( n-1 \right)$-positive (nonnegative), then the scalar curvature, Ricci curvature, and orthogonal Ricci curvature are positive (nonnegative), respectively. Second, we show that any compact Kähler Einstein manifold satisfying the condition $$\lambda_1+\dots+\lambda_{\alpha}\ge -{\alpha}\theta(n,{\alpha})\bar\lambda,\; {\alpha}\le \frac{n}{2}$$ must have nonnegative constant holomorphic sectional curvature.