Papers

We prove sharp residual stability estimates for the critical Poincaré-Sobolev equation near sums of one positive hyperbolic solution and one projected Euclidean bubble in dimensions at least four. The estimates are uniform in the relative positions of the centers. The sharp rate is linear except in dimension six, where it contains a square root of a logarithm. For nonnegative functions, the same estimates hold under an energy constraint near the sum of the two profile energies for \(\lambda\) outside a finite set, and an additional mass constraint removes this restriction. For arbitrary finite numbers of profiles, we identify and realize all nonnegative Palais-Smale levels. We characterize the parameters at which different pairs of profile numbers have the same energy and show that the limiting energy and mass determine both profile numbers.

We propose k-Hessian equations as auxiliary equations for degenera- tions of Kähler classes to nef classes of numerical dimension k. As proof-of-concept results, we provide a uniform mean-value inequality and uniform à priori estimates for Hessian equa- tions.

It is known that there can be at most 3 disjoint nontrivial minimal graphs in R^3 with 0 boundary values, and conjectured that there can be at most 2. We explore properties that the graphs would have if there were 3.

We show that, in ambient dimension at most seven, varifold limits of the minimizing movements scheme introduced by Almgren–Taylor–Wang and Luckhaus–Sturzenhecker are integral Brakke flows. As a consequence, we show that $L^1$-limit points of the scheme, usually known as flat flows, are generalized BV flows. The result is unconditional and, in particular, remains valid even in the absence of energy convergence and in the presence of fattening.

We complete the classification of hypersurfaces with constant sectional curvature of dimension greater than or equal to three in the product of two space forms.

We study spectral Gårding elliptic equations on compact Kähler manifolds. Smooth solvability is equivalent to the existence of a smooth representative in the admissible domain. The concavity theorem for ideal Gårding polynomials gives a fixed concave formulation to which Székelyhidi's second-order argument applies in every degree $2\le d\le n$, extending Lin's top-degree solvability theorem. Combined with the numerical criteria of Fu–Li–Zhang–Zhang and Chen–Nie–Xu, this yields a Nakai–Moishezon type criterion for solvability on compact Kähler manifolds. A solution converts the local ideal Gårding structure into a global Lorentzian structure. Its mixed derivatives give a strict nonlinear Hodge–Riemann hierarchy from the degree of the equation down to degree two, and the homogenization of the intersection polynomial is Lorentzian in nef class directions. Numerical criteria then produce explicit Hodge–Riemann classes, Lefschetz isomorphisms in bidegree $(1,1)$, and weighted Hodge-index inequalities.

In this note, we prove that on a compact Hermitian manifold with negative Gauduchon degree, the Chern-Yamabe flow introduced in our recent work [13] converges to the solution of prescribed Chern scalar curvature problem, provided that the candidate curvature function is nonpositive and not identically zero.

We establish sharp Alexandrov–Fenchel inequalities for closed, embedded, axisymmetric hypersurfaces Sigma in n+1 dimensional Euclidean space R^{n+1}, the dimension n > or = 2, with admissible C^{1,1} profiles. Let H_j be the j-th normalized mean curvature. If Sigma is axiconvex, i.e., every hyperplane orthogonal to the symmetry axis meets the enclosed domain in an n-dimensional ball or empty, then \[\left(\frac{\int_\Sigma H_\ell\,dA}{|\mathbb S^n|}\right)^{1/(n-\ell)} \le \left(\frac{\int_\Sigma H_k\,dA}{|\mathbb S^n|}\right)^{1/(n-k)}, \qquad 0\le\ell<k\le n-1. \] Without axiconvexity, the inequalities hold for sign-corrected integrals \[\int_\Sigma\varepsilon_\theta^{\,j}H_j\,dA,\] where varepsilon_{theta}= {sgn}(sin {theta}) denotes the sign of the azimuthal principal curvatures. As applications, the Alexandrov–Fenchel inequalities remain valid whenever both l and k are even. Equality in each of the above inequalities holds if and only if Sigma is a round sphere. The proof adapts the rearrangement and tangent angle folding approach of Dalphin-Henrot-Masnou-Takahashi from surfaces of revolution in R^3 to n-dimensional axisymmetric hypersurfaces in R^{n+1}. The higher-dimensional construction uses weighted monotone rearrangement to produce convex comparison bodies with C^{1,1} boundaries.

In this paper, we determine the structure of the cut locus for a class of complete warped product manifolds $M=(R^1\times S^1,dt^2+m(t)^2d\theta^2)$, where the warping function $m(t)$ is positive and even, under the assumption that there exists a positive number $t_1$ such that the Gaussian curvature is nonpositive on the domain $t^{-1}\left((t_1,\infty) \right)$ and is decreasing along the subarc $t^{-1}([0,t_1])\cap\theta^{-1}(0)$ of the upper half meridian.

We study complete K$\"a$hler metrics with prescribed signs of holomorphic bisectional curvature on total spaces of holomorphic vector bundles over compact complex manifolds. We establish necessary and sufficient conditions in the nonnegative and strictly negative cases, characterized respectively by unitary flatness and Griffiths negativity of the bundle, together with the corresponding curvature conditions on the base. We also obtain a global metric splitting in the nonnegative case, rule out strictly positive curvature over positive-dimensional bases, and construct counterexamples to the converse in the nonpositive case.

In this paper, we study hypersurfaces $M^4$ in the Minkowski space $\mathbb{E}^5_1$ whose shape operator $S$ is non-diagonalizable, having a triple real eigenvalue $\lambda$ with a one-dimensional eigenspace together with a simple eigenvalue $\mu$ (Case~(iii) in the classification of Magid). Assuming that $\lambda$ and $\mu$ satisfy a functional relation $\lambda=f(\mu)$ and that the eigenvector field of $\mu$ is geodesic, we show that $\lambda$ and $\mu$, as well as the pseudo-orthonormal frame, are intrinsic invariants of $M$, determined by the induced metric up to sign. As a consequence, we prove that such hypersurfaces are unique up to isometries of $\mathbb{E}^5_1$. We also obtain a classification of the warped product hypersurfaces of this type.

Let $H$ be a closed subgroup of a compact connected Lie group $G$ such that $G/H$ is simply connected, the action of $G$ on $G/H$ is almost effective, and $G/H$ is a strongly isotropy irreducible space. The smallest positive eigenvalue of the Laplace-Beltrami operator $\Delta$ associated to the standard metric on $G/H$ was recently computed. In this article we obtain a second non-zero eigenvalue of $\Delta$. As a consequence, we prove that all eigenspaces of the Laplacian are irreducible as $G$-modules if and only if $(G,H)$ is a compact symmetric pair of real rank one, or $(G,H)$ is equal to $(\operatorname{G}_2,\operatorname{SU}(3))$ or $(\operatorname{Spin}(7),\operatorname{G}_2)$.

We prove that, in every dimension, a bounded, non-constant entire solution of the Allen–Cahn equation whose zero set is contained in a half-space is a one-dimensional heteroclinic solution.

In this follow-up to \cite{Sua25}, we extend the results to the coupled anomaly flow which incorporates the evolution of a Hermitian metric on a holomorphic vector bundle over a compact complex $3$-fold with general slope parameter $\alpha'$. This yields a similar sufficient smallness condition on the slope parameter $\alpha'$ in order to extend the flow on $[0, \tau)$ to $[0, \tau + \epsilon)$. We also augment the argument of \cite{Sua25} slightly to fix the peculiarity of the units in the condition for $\alpha'$ for the $k = 1$ case.

We study the exponential Green energies associated with a finite point configuration on a Riemann surface. We focus on studying optimal or close-to-optimal configurations, and show that they must be well-spread out on the surface. We study in particular the round sphere and obtain values for the minimal Green energy of a configuration. The first version of this paper was written with the help of ChatGPT.

In this paper, we prove a global $C^{2,\alpha}$ estimate for the complex Monge–Ampère equation with a positive $C^\alpha$ right-hand side on a compact Kähler manifold.

We compute cohomology spaces of the ortosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$ on the (1,1)-dimensional real superspace with coefficients in the superspace $\frak{D}_{{\lambda},\mu}$ of $n$-ary differential operators acting on weighted densities (${\lambda}\in\mathbb{R}^n$ and $\mu\in\mathbb{R}$).

We study closed simply connected six-manifolds with positive sectional curvature and an effective isometric action of $SO(3)$ or $SU(2)$. Starting from Liu's corrected orbit-space analysis, we refine the stratification and recover smooth topology in several previously unresolved cases. For a boundaryless $SO(3)$ quotient, singular isotropy is connected, the weighted nonprincipal graph belongs to nine necessary families, and every action with only cyclic exceptional isotropy is classified equivariantly. The remaining branching families admit explicit rational intersection and Pontryagin formulas. For a three-ball $SO(3)$ quotient, the uniform-boundary configurations give sphere bundles over $S^4$ and a completely distinguished odd-integer family of non-spin six-manifolds. A doubled-quotient argument excludes the configuration with two boundary fixed points and an interior singular orbit, while the configuration with one fixed point and an exceptional interval has underlying manifold $\mathbb{CP}^3$. For a boundaryless $SU(2)$ quotient, an adjoint-weight matching argument reduces the exceptional three-orbit case to a unique action on the Eschenburg flag and identifies every two-orbit candidate with a weighted action on $S^2\times S^4$. These classifications imply rational ellipticity throughout the stated symmetry class. We also establish restricted curvature obstructions for the two diagonal product actions, without asserting nonexistence for arbitrary invariant metrics. Abstract graph restrictions, smooth realizability, and positive-curvature realizability are kept distinct. Some of the results in this paper are discovered by ChatGPT-6-astra and verified by the authors.

For every $m\geq2$, we prove that any connected closed oriented smooth immersion $X:M^m\to\mathbb{R}^{m+1}$ satisfying \[ K=\left\langle X,\nu\right\rangle \] is an embedding onto the unit sphere. No convexity is assumed. At a point maximizing $|X|$, we find a strictly convex component of $\{K>0\}$. If this component does not coincide with $M$, its polar immersion bounds an unbounded convex set. A finite area lemma then extends the local surface area formula for the polar body to a global measure identity. Using the vanishing first moment of surface area measure, we show that the origin lies in the interior of the polar body, contradicting the existence of recession directions. Thus the hypersurface is strictly convex, and the classification of strictly convex shrinkers completes the proof.

For an odd prime $p$, we prove a Smith inequality over $\mathbb{F}_p$ for exact Lagrangian Floer homology under a symplectic $\mathbb{Z}_p$-action, assuming an equivariant stable normal trivialization compatible with the Lagrangians and the required relative Pin data. As an application, we obtain a rank inequality between the symplectic Khovanov homology of a $p$-periodic link and that of its quotient. These results extend the involutive localization of Seidel and Smith. The main construction is a stabilized chain complex. At every invariant intersection point we adjoin the Morse complex of a finite-dimensional normal spectral sphere, whose generators lie in free $\mathbb{Z}_p$-orbits. The differential counts holomorphic strips coupled to sphere half-trajectories by a one-sided weighted asymptotic evaluation. We prove transversality, a uniform spectral cutoff, and the compactification and gluing statements needed for the chain identity. The stabilized complex is quasi-isomorphic to the ordinary Floer complex and contains the complex of the fixed data as a subcomplex with free quotient, so the algebraic Smith inequality applies.

This Master's thesis develops and applies the multisymplectic field-theory framework to (pure) Yang-Mills theory. Building from the geometric foundations of jet bundles and the Poincaré-Cartan forms, the work constructs the Lagrangian multisymplectic formalism for first-order field theories, recovering the usual Euler-Lagrange field equations and extending the Poisson brackets of symplectic mechanics to a full graded Lie algebra. It then establishes the Legendre correspondence with the multisymplectic Hamiltonian formalism (in which Hamilton equations are retrieved) including the case of singular Lagrangians (where a constraint algorithm is reproduced). Breaking Lorentz covariance via a Cauchy surface slicing, the thesis then relates the covariant multisymplectic picture to an instantaneous presymplectic (infinite-dimensional) formulation in which the instantaneous De Donder equations and presymplectic brackets are obtained in a compatible way. A comprehensive treatment of symmetries then leads to multisymplectic and instantaneous momentum maps generalizations and to several versions of Noether's theorems. Specializing the formalism to principal-bundle gauge theory, the thesis identifies the multisymplectic configuration space with the bundle of connections, gauge symmetries with symmetries in the multisymplectic language and builds the Yang-Mills Lagrangian. Lastly, among many computations, the standard tensorial Yang-Mills equations and the Bianchi identity in local trivializations are retrieved, Yang-Mills conserved currents and charges are obtained and the graded algebra is derived for a special class of Hamiltonian forms.

The interplay between a weighted volume measure and its naturally associated weighted Laplace operator is fundamental to the geometric analysis of weighted structures governed by gradient vector fields. Beyond the gradient setting, however, the theory has remained considerably less developed, largely because this natural correspondence is lost. In this paper, we identify invariant measures as the appropriate framework for extending the theory to general drifted Laplace operators. This approach yields a systematic treatment of parabolicity in the non-gradient setting, reveals its connection with volume growth, and leads to new volume finiteness results for non-gradient geometric structures.

We prove that Choe's higher-dimensional Enneper hypersurfaces in $\mathbb R^{n+1}$, $3\le n\le6$, cannot be confined between two parallel hyperplanes, disproving his slab confinement conjecture. We further strengthen this conclusion by determining their precise height growth exponent. If $H(R)$ denotes the maximal absolute height above the asymptotic hyperplane within the horizontal ball of radius $R$, then $H(R)=R^{\alpha_1+o(1)}$ as $R\to\infty$, where $\alpha_1=\frac{\sqrt{(n-2)^2+4\lambda_1(\Omega)}-(n-2)}{2}$, $\Omega=S^{n-1}\setminus[0,\infty)^n$, and $\lambda_1(\Omega)$ is the first Dirichlet eigenvalue of the spherical Laplacian. In particular, $0<\alpha_1<1$, so the height is unbounded but grows sublinearly. The proof combines the blow-down behavior of a fundamental piece of Choe's hypersurface with the rigidity of positive harmonic functions on cones. In dimension two, the formula gives $2/3$, recovering the classical Enneper height-growth exponent.

We classify graphs of complex polynomials in one variable up to isometry, both for the inner (length) distance and for the outer (Euclidean) distance of $\mathbb{C}^2\cong\mathbb{R}^4$. For polynomials $p,q$ with $\deg p\ge 2$ we show that the graphs of $p$ and $q$ are inner isometric if and only if they are outer isometric, if and only if $p(z)=\alpha\,q(\beta z+h)+k$ or $p(z)=\alpha\,\bar q(\beta z+h)+k$ for some $\alpha,\beta,h,k\in\mathbb{C}$ with $|\alpha|=|\beta|=1$, where $\bar q$ denotes the polynomial with conjugate coefficients. Moreover, every inner isometry between such graphs is the restriction of an isometry of $\mathbb{C}^2$; in this sense graphs of complex polynomials are isometrically rigid. We contrast this with the real case, where the two distances behave very differently: the graphs of any two real polynomials are inner isometric, whereas, for degree at least $2$, outer isometries are generated by translations, reflections in horizontal and vertical lines and half-turns.

Let $ \operatorname{MaxCol}(n,d) $ denote the maximal number of nonsimultaneous elastic collisions among $n$ identical hard balls moving freely in $\mathbb{R}^d$. We give a planar construction with exponential lower bound with respect to $n$: $ \operatorname{MaxCol}(n,2)\geq 2^{\lfloor n/2\rfloor}, \quad n\geq3 $.

For $1\leq r\leq n-1$ we study shrinking self-similar ends of the $r$-mean curvature flow, $S_r=c\langle X,N\rangle$ with $c<0$, that are asymptotic to a cone. For every rotational cone of positive slope we construct a smooth embedded rotational end and prove that its profile is the only positive solution of the profile equation with the prescribed slope at infinity. Its angular defect $xu'-u$ is strictly negative, which yields $r$-admissibility, and the profile has a complete asymptotic expansion for $u(x)/x$ in powers of $x^{-(r+1)}$, valid after arbitrary differentiation. The profiles depend real analytically on the slope and, on sufficiently distant exterior regions, foliate the image of the corresponding family. Our main result removes the symmetry assumption: two ends that converge smoothly, with multiplicity one, to the same circular cone coincide outside a compact set. The proof transforms the equation for their difference into an almost-eigenfunction inequality for a drift Laplacian on an auxiliary conical metric and applies Bernstein's asymptotic analysis on weakly conical ends. The argument extends to strictly $r$-admissible cones whose links have constant principal curvatures.

We construct a symmetric trace-free tensor for constant-mean-curvature hypersurfaces in $\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$ that extends the Abresch-Rosenberg tensor. Within its natural class, it is characterized by the linearity of its Codazzi defect, and its complete connected zero set consists exactly of the rotational CMC spheres. An exact integrated Simons identity yields explicit gap theorems, valid for every $h>1$ in $\mathbb{H}^n\times\mathbb{R}$ and every $h>(n\int_0^{\pi/2}\sin^{n-1}r\,dr)^{-1}$ in $\mathbb{S}^n\times\mathbb{R}$. We compare the pinching thresholds with the pointwise algebraic level that implies convexity. In $\mathbb{H}^n\times\mathbb{R}$ the explicit pinching conditions imply strict convexity, so there the method gives an independent proof with explicit constants. In $\mathbb{S}^n\times\mathbb{R}$ the pinching region exceeds the convexity level for $n\in\{3,4\}$ and every admissible $h$, and for $n\geq5$ whenever $h^2\geq n/(6(n+2))$; there the gap theorem does not require convexity to be known in advance. We also determine second-order large-$h$ asymptotics of the thresholds.

In every sub-Riemannian Heisenberg group $\mathbb H^n$, with $n\geq 2$, we prove the existence of a global perimeter minimizer whose boundary is a smooth, non-characteristic, entire intrinsic graph but it is not a vertical hyperplane.

We formalize the Hamilton–Perelman Ricci-flow proof of Thurston's geometrization conjecture. The public Lean declaration is reproduced in this paper. Its geometric interpretation follows, and verification is recorded in 4.

We construct a smooth embedded annulus in $R^3$ with negative Gauss curvature in its interior whose boundary consists of two strictly convex curves in planes tangent to the annulus, and which contains two closed asymptotic curves. This gives a counterexample to a 1962 conjecture of Nirenberg. The construction is based on Lelieuvre's formula, which yields a closed curve that is then realized as an asymptotic curve on a ruled annulus. The second asymptotic curve is obtained from the periodic solution of a Riccati equation.