New gravitational instantons and non-holomorphic minimal spheres discovered.
problem Existence of non-holomorphic minimal spheres in ALF spaces.
method Gluing construction of ALF gravitational instantons using specific metrics.
result Existence of non-holomorphic minimal spheres in a general class of ALF spaces.
We generalize here our general procedure for constructing constant curvature maps of 2-spheres into Grassmannian manifolds G(m,n) this time concentrating our attention on maps which are non-holomorphic. We present some expressions describing these solutions in the general case and discuss how to use these results to co…
The paper proves a disk's energy minimizer is holomorphic and calculates its Morse index.
problem Analyzing the Morse index of a non-holomorphic disk in pseudoconvex domains.
method Proof of holomorphic minimizers and Morse index calculation.
result Non-holomorphic critical disks have a Morse index of at least n-1.
Study finds conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
problem Finding conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
method Restriction of Donaldson's functional to diagonal metrics on Higgs bundles with non-holomorphic Higgs fields.
result Provides necessary and sufficient conditions for the functional to attain a minimum.
We construct two types of non-holomorphic Lefschetz fibrations over S2 with (−1)-sections ---hence, they are fiber sum indecomposable--- by giving the corresponding positive relators. One type of the two does not satisfy the slope inequality (a necessary condition for a fibration to be holomorphic) and has a simpl…
For any finitely presentable group G, we show the existence of an isolated complex surface singularity link which admits infinitely many exotic Stein fillings such that the fundamental group of each filling is isomorphic to G. We also provide an infinite family of closed exotic smooth four-manifolds with the fundam…
The study constructs minimal submanifolds in complex and quaternionic projective spaces.
problem Finding minimal submanifolds in complex and quaternionic projective spaces.
method Using complex-valued harmonic morphisms.
result Complete minimal submanifolds of odd-dimensional complex projective spaces and their dual hyperbolic spaces are constructed.
The Fubini-Study metric minimizes a volume-normalized holomorphic systole in CPn.
problem Finding metrics with minimal holomorphic systoles in complex projective spaces.
method Introduced holomorphic k-systole and used Gauduchon metrics to establish minimization. result The Fubini-Study metric locally minimizes the volume-normalized holomorphic (n−1)-systole. Study minimal surfaces in 4D, find specific tori with total curvature -8π.
problem Find complete proper non-holomorphic minimal tori in R^4 with total curvature -8π.
method Use tools like Gauss maps and link/braid/writhe at infinity. Translate problem into a system of equations involving Weierstrass function. Explicitly solve for rectangular and square tori.
result Explicit solutions for minimal tori in R^4, including generalization of Chen-Gackstetter torus in R^3.
Minimal Kaehler submanifolds up to codimension four are studied.
problem Characterizing Kaehler submanifolds in high codimensions.
method Analyzing isometric immersions and compositions of submanifolds.
result Holomorphicity or composition of submanifolds with specific properties.
The paper confirms a conjecture about submanifolds in Euclidean space.
problem Addressing a conjecture about non-holomorphic Kaehler submanifolds in Euclidean space.
method Analyzing the structure of the second fundamental form and ruling dimensions.
result The conjecture is confirmed for codimensions p ≤ 6, and for p = 7 to 11 under additional assumptions.
In this note we prove a Weierstrass representation formula for pluriminimal submanifolds of euclidean spaces. We use this formula to produce new families of examples of pluriminimal submanifolds. We also prove that any affine algebraic manifold can be pluriminimally embedded into some euclidean space in a non holomorph…
Constant curvature surfaces are constructed from the finite action solutions of the supersymmetric CPN−1 sigma model. It is shown that there is a unique holomorphic solution which leads to constant curvature surfaces: the generalized Veronese curve. We give a general criterion to construct non-holomorphic…
We show how certain stabilizations produce infinitely many closed oriented 4-manifolds which are the total spaces of genus g surface bundles (resp. Lefschetz fibrations) over genus h surfaces and have non-zero signature, but do not admit complex structures with either orientations, for "most" (resp. all) possible value…
This paper gives a construction for all minimal immersions f of the Poincaré disc into the complex hyperbolic plane CH2 which are equivariant with respect to an irreducible representation ρ of a hyperbolic surface group into PU(2,1). We exploit the fact that each such immersion is a twisted conformal …
A new approach for the construction of finite action solutions of the supersymmetric CPN−1 sigma model is presented. We show that this approach produces more non-holomorphic solutions than those obtained in previous approaches. We study the CP2 model in detail and present its solutions in an e…
Study on spheres with minimal equators.
problem Classifying metrics on spheres with minimal equators.
method Survey and discussion of related problems.
result Classification of metrics on spheres with minimal equators.
The 3-sphere has either 2 minimal 2-spheres or an optimal foliation by 2-spheres.
problem Proving existence of minimal 2-spheres or optimal foliations in arbitrary Riemannian 3-spheres.
method Analyzing the properties of arbitrary Riemannian metrics on 3-spheres.
result The existence of at least two minimal 2-spheres or an optimal foliation in 3-spheres with arbitrary metrics.
The paper explores complex Poisson structures on smooth functions in complex manifolds.
problem Exploring complex Poisson structures on smooth functions in complex manifolds.
method Considering structures of complex Poisson brackets generated by a (1,1)-form. result Examples of complex Poisson structures are provided in $\C^\ast$.
Disk complexes show 3-sphere surfaces are topologically minimal.
problem Understanding minimal surfaces in 3-sphere topology.
method Analyzing disk complexes of genus >1 Heegaard surfaces.
result Genus >1 Heegaard surfaces have minimal index 2g-1.
Proves existence of at least two minimal spheres in any 3D space.
problem Existence of minimal spheres in arbitrary 3D spaces.
method Iterative relative min-max constructions.
result Proves existence of at least two embedded minimal spheres.
Proves optimal regularity for sphere minimizers in 3-sphere.
problem Finding optimal regularity for sphere minimizers.
method Proves C1,1 regularity for minimizers of prescribed mean curvature over isotopy classes. result Proves optimal C1,1 regularity for minimizers. Minimal spheres found in ellipsoids with large axes.
problem Finding non-planar minimal spheres in ellipsoids.
method Quantitative proof of minimal spheres existence with constraints.
result Ellipsoids with large axes contain at least three non-planar minimal spheres.
Minimal submanifolds are stable in certain conformal spheres.
problem Stability of minimal submanifolds in conformal spheres.
method Analyzing n-dimensional Riemannian spheres with specific curvature conditions. result Closed stable minimal submanifolds are not found in δ-pinched conformal spheres. Four minimal spheres found in sphere with special metric.
problem Existence of minimal spheres in spheres with specific metrics.
method Simon-Smith min-max theory for multiplicity one theorem.
result At least four embedded minimal 2-spheres proven.
Minimal surfaces in spheres found for any genus.
problem Finding minimal surfaces with arbitrary genus in 3-spheres.
method Topological structure analysis and embedding theorem.
result Every positive Ricci curvature 3-sphere contains a genus g surface.
We first extend the notion of connection in the context of Courant algebroids to obtain a new characterization of generalized Kaehler geometry. We then establish a new notion of isomorphism between holomorphic Poisson manifolds, which is non-holomorphic in nature. Finally we show an equivalence between certain configur…
Study on minimal hypersurfaces in a unit sphere, proving specific isometries.
problem Characterizing minimal hypersurfaces with constant scalar curvature.
method Analyzing n-dimensional complete minimal hypersurfaces in a unit sphere with constant scalar curvature. result Proves isometry to totally geodesic sphere or Clifford torus under certain conditions.
Minimal hypertori found in 4D sphere, solving Bernstein conjecture.
problem Finding minimal embedded hypertori in 4D sphere.
method Analyzing minimally embedded and immersed hypertori and hyperspheres.
result Infinitely many non-isometric minimally embedded hypertori and hyperspheres found.
5 minimal tori found in 3-spheres with positive Ricci curvature.
problem Existence of at least 5 embedded minimal tori in 3-spheres with positive Ricci curvature.
method Combination of min-max theory and mean curvature flow heuristics.
result Confirms B. White's conjecture for positive Ricci curvature.
Study finds many nonplanar minimal spheres in elongated ellipsoids.
problem Existence of nonplanar minimal spheres in elongated ellipsoids.
method Global bifurcation techniques to establish existence and quantify number.
result Arbitrarily many nonplanar minimal spheres exist in elongated ellipsoids.
In this note we complete the discussion of minimality of symplectic fiber sums. We find, that for fiber sums along spheres the minimality of the sum is determined by the cases discussed by M. Usher and one additional case: If the sum is the result of the rational blow-down of a symplectic -4-sphere in X, then it is non…
New minimal surfaces found in spheres and hyperbolic spaces.
problem Constructing minimal submanifolds in even-dimensional spheres and hyperbolic spaces.
method Using complex-valued harmonic morphisms.
result Explicit examples of minimal submanifolds in S4 and H4. The study of stable and index compact minimal submanifolds in Berger spheres.
problem Stability and index of compact minimal submanifolds in Berger spheres.
method Analyzing stability and index properties of compact minimal submanifolds in Berger spheres.
result Stable compact minimal submanifolds exist in Berger spheres for specific values of τ, and their classification is provided.
Minimal surfaces with negative curvature found in large spheres.
problem Existence of minimal surfaces with negative curvature in large dimensional spheres.
method Applied Song's strategy to closed Riemann surfaces with large automorphism groups, resulting in almost hyperbolic minimal surfaces.
result Existence of closed minimal surfaces with negative induced curvature in any sphere of large dimension.
The paper constructs metrics on spheres with families of minimal hypersurfaces.
problem Finding Riemannian metrics on spheres with specific families of minimal hypersurfaces.
method Used Nash-Moser Inverse Function Theorem in the tame maps setting.
result Generalized Guillemin's theorem for Zoll families of minimal hypersurfaces.
The paper proves the existence of at least 4 embedded minimal tori in a three-sphere with positive Ricci curvature.
problem Proving the existence of embedded minimal tori in three-spheres with positive Ricci curvature.
method The proof relies on a multiplicity one theorem for the Simon-Smith min-max theory.
result There exist at least 4 distinct embedded minimal tori in the three-sphere with positive Ricci curvature.
A minimal hypersurface in a sphere is uniquely determined.
problem Characterizing closed minimal hypersurfaces in spheres.
method Proving strong rigidity of closed minimal hypersurfaces.
result Closed minimal hypersurfaces in spheres are uniquely determined.
Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the product of two spheres is obtained. Also, it is proved that the only stable comp…
Study finds bound on energy of minimal spheres on complex manifolds.
problem Finding bounds on energy of minimal spheres on complex manifolds.
method Proving existence of harmonic spheres with Morse index bound one.
result Sum of energies of minimal spheres realizes a geometric invariant width.
Minimal surfaces reflect across spheres, proving annulus uniqueness.
problem Uniqueness of free boundary minimal annuli in balls.
method Reflection principle applied to minimal surfaces meeting spheres at 90 degrees.
result Every embedded free boundary minimal annulus in a ball is the critical catenoid.
The study finds conditions for area-minimizing cones over submanifolds.
problem Conditions for area-minimizing cones over submanifolds.
method General configuration results for area-minimizing cones.
result Cone over the minimal product of submanifolds and spheres are area-minimizing.
New formulas for minimal surfaces with specific end conditions.
problem Existence and explicit formulas for minimal surfaces with embedded planar ends.
method Provided new explicit formulas for genus 0 minimal surfaces in R^3 with 2k+1 embedded planar ends.
result Existence and explicit formulas for minimal surfaces with 2k+1 embedded planar ends for all k ≥ 4.
Study finds the volume of unit vector fields on a punctured sphere and shows their images match minimally immersed Klein bottles.
problem Finding the volume of unit vector fields on a punctured sphere.
method Analyzes the volume of unit vector fields on an antipodally punctured unit 2-sphere and shows their images coincide with minimally immersed Klein bottles.
result The images of minimizing vector fields on the punctured sphere match those of minimally immersed Klein bottles.
The paper examines deformations of pseudoholomorphic curves in a nearly Kähler sphere.
problem Investigating rigidity and deformability of pseudoholomorphic curves in S6. method Analyzing moduli space of minimal surfaces isometric to pseudoholomorphic curves.
result Describes the moduli space of noncongruent minimal surfaces isometric to pseudoholomorphic curves.
New Zoll families of minimal spheres found in spheres and projective spaces.
problem Finding new Zoll families of minimal spheres in various spaces.
method Equivariant constructions using Nash-Moser-Hamilton implicit function theorem.
result First examples of metrics on real projective spaces with Zoll families of minimal projective hyperplanes.
New findings on minimal isometric immersions of flat n-tori into spheres.
problem Conditions for minimal isometric immersions of flat n-tori into spheres.
method Analyzes rationality conditions and derives upper bounds for algebraic irrationality degree.
result Upper bound for algebraic irrationality degree of minimal isometric immersions is sharp and equals 4 for n=3.
The study finds conditions for minimal spheres into ellipsoids using eigenfunctions.
problem Conditions for branched minimal immersions of spheres into ellipsoids to be embedded.
method Using eigenfunctions with respect to a critical metric, the study provides sufficient conditions for embeddings and constructions of non-planar minimal spheres.
result Conditions for embeddings and constructions of non-planar minimal spheres using eigenfunctions.