Neural training recovers Willmore minimizers for genus 0 and 1
A PINN-style loss on neural embeddings yields the round sphere and Clifford torus while searching for genus-2 candidates.
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Differential Geometry
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis
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Round disk demands the greatest distortion in map projections
Every equal-area convex region fits in a plane map with distortion at most rho/sin rho.
All (p,q)-pinwheel embeddings in B_{p,q} are Hamiltonian isotopic
The nearby Lagrangian conjecture holds for these singular Lagrangians because the symplectomorphism group is generated by a single twist.
Analyticity sets exact uniqueness threshold in Calderón problems
Uniqueness holds for analytic metrics but fails densely in every non-analytic Gevrey class for both fixed-potential and fixed-frequency data
· “A Sharp Regularity Threshold for Uniqueness in Riemannian Calder\'on-type Problems”
Coarea formula holds for maps from Heisenberg group to R^{2n}
A new integral for the area of half-Holder curves makes this work even for the simplest vector-valued case.
· “Area of H\"older curves and coarea formula on the Heisenberg group”
One weighted inequality identifies the strictest minimal cones
A distance-weighted perimeter gap is equivalent to strict stability plus strict minimality.
Hamiltonian stationary graphs: singular set has dimension at most n-5
Every weak solution is smooth off a small singular set, and a new 5-D cone shows the bound is optimal.
· “Sharp partial regularity of Hamiltonian stationary and special Lagrangian graphs”
Böhm metrics gain infinitely many unstable modes
The number of negative Lichnerowicz modes on each Böhm sequence diverges, signalling unstable black holes.
In 3D and 4D, string-background solitons are sphere quotients
Rigidity results also separate strong-torsion string backgrounds from special-holonomy spaces.
· “Topology of low-dimensional generalized Ricci solitons and string backgrounds”
Vortex surfaces as coordinates tame slender-vortex motion
The framework recovers the classic ring-speed formula with axial flow and solves for the deformed vorticity surfaces.
· “Vortex filament dynamics and vortex ring motion revisited”
Log-minor coordinates certify proper matrix actions
Two explicit coordinate-map tests decide when a subgroup acts properly on an upper-triangular homogeneous space.
· “Proper Actions on Homogeneous Spaces of the Upper Triangular Matrix Group via Coarse Geometry”
Every nonorientable surface embeds minimally in S^4
Area below 8π unlocks Willmore minimizers of every topology in R^4.
· “Nonorientable Minimal Surfaces Embedded in the Round 4-sphere”
Kähler parabolic Higgs sheaves obey the Bogomolov–Gieseker inequality
Stability forces a nonnegative discriminant; vanishing Chern numbers yield a pluriharmonic metric on the divisor complement.
A partial sign on the Calabi curvature operator forces CP^n or flat
A new Bochner formula turns one partial curvature sign into full rigidity of Kähler-Einstein spaces.
· “A characterization of complex projective space via the Calabi curvature operator”
A Liouville theorem produces Evans-Krylov regularity by blowup
For the complex Monge-Ampère equation, the method works even where classical C^3 estimates fail, on cylinders and cones.
J-equation breakdown set equals J-non-ample locus
Under J-bigness and a boundary cone condition, the degeneracy locus is a finite union of prime divisors and one current realizes it.
· “The J-equation on K\"ahler manifolds under a smooth boundary cone condition”
A singular continuity theorem ties alpha and delta invariants
The equality also pins the critical exponent for positively curved KE metrics to the normalized volume.
· “Demailly-Koll\'ar continuity on klt pairs, and applications to alpha and delta invariants”
Mixed-polarization quantum states converge to Bohr–Sommerfeld states
At large flow time, mixed-polarization quantum states match the Bohr–Sommerfeld basis indexed by lattice points
· “Limits of quantization from mixed to real polarizations on toric varieties”
This paper classifies positively curved spaces with a strong fixed-point symmetry action
Fixed point homogeneous positively curved orbifolds are, up to equivariant diffeomorphism, spheres, the Cayley plane, or weighted complex…
· “Fixed Point Homogeneous Orbifolds with Positive Sectional Curvature”
No equilibrium for multiple rotating black holes
Every inter-horizon axis rod attracts, so stationary vacuum spacetimes have a single black hole.
A symplectic-Hessian bound forces entire solutions to be quadratic
Pointwise curvature estimate extends Bernstein rigidity and sharpens the planar constant to 3.
One derivation generates all four flight-control inversion laws
A unified equation yields model-based, sensory, incremental, and hybrid NDI laws and shows how to choose between them.
· “Dynamic Inversion: An Incrementally Evolving Methodology for Flight Control Design”
Escobar's eigenvalue conjecture fails in every dimension
Conformal ball metrics have positive Ricci and convex boundary, yet their first Steklov eigenvalue dips below 1.
A new theorem shows that in a curved space with a carefully balanced weight function
In three-dimensional metric measure spaces satisfying a specific curvature-weight condition, compact genus-zero surfaces with constant…
· “Hopf type theorem for surfaces of constant weighted mean curvature in Riemannian manifolds”
This paper proves that a complete Riemannian manifold with nonnegative scalar curvature…
In dimensions 3 to 7, the mass of the distinguished asymptotically flat end of a half-space manifold with nonnegative scalar and boundary…
· “Positive Mass Theorem with Arbitrary Ends and Noncompact Boundary”
EH maps from flags to quadrics factor through embeddings
A Penrose-type transform identifies their moduli space with M/S^1 and preserves minimal mean-curvature norm.
· “Transforms of holomorphic maps from flag manifolds into Grassmannians”
This paper proves a priori interior curvature estimates and smoothness for solutions of…
A priori interior curvature estimates and C^1-to-smooth regularity hold for the scalar curvature equation in dimension 4.
· “Interior estimates and regularity for the scalar curvature equation in dimension 4”
This paper proves that for a three-dimensional Kähler manifold with a J-semistable pair…
On any compact Kähler threefold, a J-semistable pair has a J-null locus that is a finite union of curves and divisors, and the J-flow…
Wishart laws form a tensor-product category
The paper shows covariance matrices combine block-wise as tensor products, with the Wishart density as a unique geometric soliton.
· “Wishart Matrices and Quantum Geometry: Foundations and Applications in Quantum Information”
Polyhedral surfaces get a discrete uniformization theorem
Every hyperbolic polyhedral surface maps to a standard model; Euclidean and spherical cases need one bound.
Four-forms build Ricci-flat string algebroids
SU(m), G2, Spin(7), and Sp(k) geometries all carry generalized Ricci-flat metrics with torsion.
Triangular loop holonomy defines graph curvature
Each edge carries metric-compatible transport; loop products become curvature and drive SPD-preserving metric updates.
· “Geometric Structures on Graphs: a Holonomy-Based Discretization of Curvature”
Blow-up metrics can match base scalar curvature to any tolerance
In particular, positive scalar curvature survives blow-ups along complex submanifolds.
· “Scalar curvature of blow-ups of compact K\"ahler manifolds along complex submanifolds”
Fibered 5-manifold boundaries extend to relative 4-sections
A higher-dimensional analogue of Heegaard splittings now reaches 5-manifolds, with diagrams and gluing.
· “Relative multisections of higher-dimensional manifolds with boundary”
Length penalty yields infinite lemniscates
At Fresnel phases 3π/4 and 7π/4 the quarter-arcs glue smoothly; waves split off from circles.
Mass with a vector field equals ADM mass plus a nonnegative defect
This decomposition proves positivity and rigidity in every dimension and yields new charged mass inequalities.
Steklov spectra cannot drift: planar isospectral families are compact
A sequence of smooth planar domains with identical Steklov spectra always has a smoothly converging subsequence, settling a long-open…
· “The Steklov Determinant and Compactness of Isospectral Planar Domains”
Willmore CMC hypersurfaces in S^5(1) are all isoparametric
Constant scalar curvature forces every closed example into four explicit shapes: spheres, Clifford tori, Cartan hypersurfaces.
· “On the Rigidity of Closed CMC Hypersurfaces in mathbb{S}⁵(1) with Constant Scalar Curvature”
Stable minimal 3-folds in positive curvature are forced rigid
No weak bounded-geometry assumption: a uniform positive scalar gap alone forces the hypersurface to be totally geodesic.
· “Stable Minimal Hypersurfaces in Positively Curved 4-Manifolds”
Non-even Lamé equations are classical Lamé in disguise
A single parameter shift transfers monodromy, spectra, and finite-gap results across the whole component.
· “Componentwise Geometry and Monodromy of Generalized Lam\'e Equations”
Only origin-centered spheres are stable in anti-Gaussian space
The proof shows origin-centered balls are the only local minimizers of the anti-Gaussian isoperimetric problem.
· “Stable constant weighted mean curvature hypersurfaces in anti-Gaussian space”
Volume equality in Bray's bound forces a round sphere
Equality in the Jiang-Li-Wang volume bound occurs only for the round sphere of radius (1+ε)^(-1/2).
Circle-bundle trick gives positive curvature on S2 x S3
An explicit perturbed connection metric kills all flat planes via a Gauss-equation term from shrinking circle fibers.
· “A Metric with Positive Sectional Curvature on S²times S³”
A single phase yields a continuum of solutions
Same continuous phase and boundary data admit many distinct continuous viscosity solutions in every dimension n≥2.
· “Nonuniqueness of solutions to the Lagrangian mean curvature equation”
Same nullcline identity links Hopf location and criticality
A five-term trace-field formula gives the first Lyapunov coefficient in predator-prey systems.
Two-to-one bubble weights yield explicit topology at infinity
Interior bubbles cost twice boundary bubbles; homology of the resulting spaces is computed exactly.
Nilpotent nilsoliton torsion is always harmonic
For a generalized nilsoliton, the preferred splitting's three-form is both closed and coclosed; this was open.
· “Harmonic torsion of generalized nilsolitons on nilpotent Lie groups”
First nonhomogeneous Kähler–Einstein metrics embed into CH^∞
First complete nonhomogeneous Kähler–Einstein examples induced by CH^∞, with flat and Sasakian consequences.
· “Complex Hyperbolic Immersions of Cheng--Yau Metrics on Thullen Domains”
Surface diffusion can pinch off a torus in finite time
The flow stays embedded until one point blows up, shaped by a universal conical profile.
Two-sided Ricci bounds found for statistical submersions
A Hineva-type lower estimate joins the known Chen-Ricci upper estimate to control vertical curvature from both sides.
A Ricci symmetry forces rigidity in five space families
Gradient shrinkers, quasi-Einstein, static, CPE, and Miao-Tam metrics all gain parallel Ricci tensor.
Involutive distributions on Z-graded manifolds are always integrable
The strong form holds: unique maximal integral submanifolds exist, not just local integrability.
· “On the Integrability of Distributions in mathbb{Z}-graded Geometry”
Stable s-minimal cones in 3D are flat for tiny s
Near s=0 every stable nonlocal minimal cone is just a half-space, completing a three-dimensional classification.
· “Stable s-minimal cones in mathbb{R}³ are flat for s close to zero”
Hamilton's theorem is now machine-proved with zero custom axioms
Every closed 3-manifold with positive Ricci curvature is a spherical space form, now verified in the Lean assistant.
· “A Lean Formalization of Hamilton's Three-Manifold Theorem”
Flux torus quivers yield new G2-holonomy spaces
For every non-center E8 flux, a circle-fibered Calabi-Yau builds a 7-manifold whose invariants match the 4d quiver.
Concave diffusivity yields Harnack bounds on curved space
A Bochner–discriminant argument plus Nash–Moser iteration controls non-power diffusion laws.
Explicit volume gap found for minimal submanifolds in spheres
The bound works in every codimension and replaces heat-kernel constants with a polynomial one.
Elliptic points yield holomorphic discs with half-order smoothness
New blow-up method lifts the regularity of disc fillings and handles higher-index complex points.
· “The Bishop family of holomorphic discs: regularity and higher index”
Macroscopic curvature forces 1-width bound in all dimensions
Complete manifolds with small balls and fillable cycles have width at most 24(b+1)t_{n-2}.
Every Ricci soliton on H²×R is expanding and classified
The soliton fields form a translate of the Killing algebra, while gradient solitons reduce to a one-parameter vertical subfamily.
Bray volume conjecture proved for all dimensions n≥3
A small Ricci lower bound plus a scalar curvature bound caps volume at the round sphere's, with equality only for the round metric.
The entropy spectrum of a hyperbolic surface is reverse well-ordered
Subsurface entropies accumulate exactly where the complement still has room for a curve, with a real gap below 1.
Constant-scalar shrinkers rigid if they resemble plane-sphere product
A Ricci lower bound on its level sets forces the shrinker to be exactly R² × Sⁿ⁻².
· “Rigidity of shrinking gradient ricci soliton with constant scalar curvature”
Each boundary function selects exactly one self-shrinker
Flow in Minkowski space is shown to converge to it, ending compact-case rigidity.
· “Inverse Hessian Curvature Flow in Minkowski Space II: The Dirichlet problem at infinity”
A stability number detects positive scalar curvature Kähler metrics
In Kähler geometry, the threshold σ>0 marks exactly the classes carrying a PSC metric; σ>s marks a unique cscK metric.
· “Positive scalar curvature K\"ahler metrics and shifted K-stability”
Positive scalar curvature Kähler metrics hinge on one equation
Existence is equivalent to solvability, coercivity, and geodesic stability; toric classes always work.
· “A Variational Characterization of Positive Scalar Curvature K\"ahler metrics”
Calabi-Yau metrics make neural descent oscillate
A constant determinant forces eigenvalue blow-ups, destroying smoothness and monotone descent.
AVR near 1 forces a metric soliton to be Gaussian
A new gap theorem also gives epsilon-regularity and rigidity for ancient Ricci flows.
Rho tensor fully determined by Weyl torsion
The full Weyl form and harmonic curvature become computable from the Weyl torsion and the Tanaka–Webster connection.
Watermark survival is set by edit placement
The same retention rate can leave half, a quarter, or none of the signal, depending on which seeding windows survive.
82.9% cut in unseen-condition error via a reusable field step
A componentwise field-collection stage also gives exact SO(3) equivariance on the sphere and reusable condition caches.
A line forces an almost Kähler cover to split off a complex line
Nonnegative Ricci curvature plus a single line makes the universal cover a product, yielding integrability and non-hyperbolicity
· “An almost K\"ahler Cheeger--Gromoll splitting theorem with applications”
Conical Kähler–Einstein limits have unique tangent cones
A log version of Donaldson–Sun theory recovers the boundary divisor from the metric and pins down the tangent cone at every point.
Nonflat tori refute a 1994 flatness question
For every n≥3, new curved tori give each primitive hypersurface class an area-minimizing foliation; stable norms can't detect flatness.
· “Minimal foliations, codimension-one stable norms, and a question of Bangert”
Any compact nowhere dense set is almost a singular set
Uniform ellipticity alone cannot prevent Sing(u)=K∪A; pointwise singular sets still have a positive dimension gap.
· “On Singular Sets of Fully Nonlinear Uniformly Elliptic Equations”
Intermediate curvature caps ball growth at m-1 dimensions
Balls grow like radius to the m-1 power below the Ricci scale; beyond it only an exponential factor appears.
· “Volume Growth under Positive Intermediate Curvature and a Ricci Lower Bound”
Monodromy is Zariski dense on every non-hyperbolic surface
New exclusion of affine monodromy closes the last case of the authors' conjecture.
· “Zariski-Dense Monodromy of Singular Hyperbolic Metrics on Non-Hyperbolic Riemann Surfaces”
Gluing yields monopoles on cylinder-like 3-manifolds
Turning singular Dirac monopoles and BPS clusters into smooth SU(2) solutions of any positive charge and large mass.
· “Bogomolny Monopoles on Asymptotically Cylindrical 3-Manifolds”
Four equations decide when an anti-self-dual space is Einstein
A curvature obstruction gives necessary and sufficient conditions, from four-dimensional metrics to Grassmannian geometries.
· “Conformally Einstein anti-self-dual spaces and their generalisations”
Cryo-EM images yield protein backbones in one step
A rotation-group gradient flow deforms a template until its projections match noisy images—no density map needed.
