We approximate derivatives of functions on manifolds by embedding them and applying vector-valued operators.
problem Derivatives of manifold-valued functions are harder to approximate than vector-valued functions.
method Embed the manifold into a higher space, approximate the derivative of the vector-valued function, and project back.
result We provide error bounds for the approximation of manifold-valued function derivatives.
The paper generalizes inequalities on almost Kähler manifolds.
problem Generalizing inequalities on almost Kähler manifolds.
method Considered Donaldson gauge functional and twisted Aubin functionals.
result Generalized inequality between Aubin functionals.
Extends Einstein-Hilbert functional definition for stable manifolds.
problem Stability of Einstein manifolds on Riemannian manifolds.
method Second variation of generalized Einstein-Hilbert functional.
result Properties of stable Einstein manifolds presented.
Defines concordance of Morse functions on manifolds and presents a condition.
problem Deciding if two Morse functions on the same manifold are concordant.
method Introduces concordance as a stronger equivalence relation than cobordism, and presents a necessary and sufficient condition for concordance.
result A necessary and sufficient condition for two Morse functions to be concordant is presented.
The abstract discusses the equivalence of transnormal and isoparametric functions on compact manifolds.
problem The existence of transnormal and isoparametric functions on compact manifolds.
method Exploring embedded transnormal systems and showing the existence of transnormal functions on Riemannian manifolds.
result Compact manifolds with transnormal functions also have isoparametric functions, and vice versa.
Smooth Busemann functions found in harmonic Finsler spaces.
problem Analyzing Busemann functions in Finsler manifolds.
method Investigation of Busemann functions in general and asymptotically harmonic Finsler manifolds.
result Smoothness of Busemann functions on asymptotically harmonic Finsler manifolds.
The study finds monotonic properties of harmonic functions on 3-manifolds with a flat end.
problem Understanding harmonic functions on 3-manifolds with specific ends.
method Derives monotonic properties of positive harmonic functions on 3-manifolds with nonnegative scalar curvature and asymptotically flat ends.
result Rigidity characterization of spatial Schwarzschild manifolds with two ends.
Characterizes smooth functions on manifolds with simple Reeb spaces.
problem Understanding the structure of Reeb spaces for smooth functions.
method Analyzes smooth functions on closed manifolds to determine their Reeb spaces' structure.
result Characterizes smooth functions whose Reeb spaces are finite graphs.
Study the relative volume function on AH manifolds and its applications.
problem Characterize the height of geodesic defining functions and capacity of balls.
method Define and analyze the relative volume function, proving its boundedness and regularity.
result Uniformly bounded relative volume function at infinity, bound dependent only on dimension.
Study families of Morse functions for manifolds with boundary.
problem Characterize degeneracies in 1-parameter families of Morse functions.
method List all possible degeneracies in generic 1-parameter families.
result Identified all degeneracies in generic 1-parameter families.
Researchers extend Godbillon-Vey functional to almost contact manifolds, finding critical structures.
problem Finding optimal almost contact manifolds using the Godbillon-Vey functional.
method Introduced a Godbillon-Vey type functional for 3D almost contact manifolds and found its Euler-Lagrange equations.
result Constructed critical 3D almost contact manifolds with double-twisted product structure.
Defines spectral Einstein functional for manifolds with boundary.
problem Calculating the spectral Einstein functional for manifolds with boundaries.
method Defined spectral Einstein functional associated with the Dirac operator and proved a theorem for 4D manifolds.
result Proof of Kastler-Kalau-Walze type theorem for spectral Einstein functional.
Smooth manifolds have functions with exactly two critical values.
problem Characterizing manifolds with specific Reeb functions.
method Proving existence of Reeb functions with prescribed critical values.
result Characterization of manifolds in dimensions 3 and n≥5 using Reeb functions.
Classifies Morse functions on 3-manifolds made from simple building blocks.
problem Classifying Morse functions on 3-dimensional manifolds.
method Examined Morse functions on 3-manifolds represented as connected sums of Heegaard genus one manifolds.
result Found conditions for the existence of Morse functions with specific properties.
We analyze critical points of two functionals of Riemannian metrics on compact manifolds with boundary. These functionals are motivated by formulae of the mass functionals of asymptotically flat and asymptotically hyperbolic manifolds.
The study establishes inequalities for functions on manifolds using Green function estimates.
problem Developing inequalities for functions on manifolds.
method Used integral representations and uniform estimates for Green functions.
result Proved Lp Sobolev-type and Poincaré-type inequalities for functions on real and complex manifolds. Study distance functions on manifolds linking geometry to topology.
problem Understanding the relationship between curvature and topology on manifolds.
method Analyzing distance functions and their connection to manifold geometry.
result Alternative proofs of theorems linking curvature and topology.
Study proposes a functional for LCK metrics on complex manifolds.
problem Existence problem for locally conformally Kähler metrics on compact complex manifolds.
method Introducing and studying a functional that varies by complex dimension.
result Functional approach to existence problem for LCK metrics.
Unified proofs of weak holomorphic Morse inequalities using Bergman kernel functions.
problem Weak holomorphic Morse inequalities on various types of manifolds.
method Asymptotic Bergman kernel functions and Bochner-Kodaira-Nakano formulas.
result Unified proofs of weak holomorphic Morse inequalities.
First example of open manifold with positive Ricci curvature and non-proper Busemann function.
problem Counterexample to Busemann function properness in open manifolds with nonnegative Ricci curvature.
method Provided an open manifold with positive Ricci curvature and non-proper Busemann function.
result First example of open manifold with positive Ricci curvature and non-proper Busemann function.
Establishes Poincaré's lemma for formal manifolds.
problem Developing smooth relative Lie algebra homologies and cohomologies.
method Theory of formal manifolds and formal Lie groups.
result Poincaré's lemma for de Rham complexes with formal functions and generalized functions.
Proves theorem for Riemannian manifolds, extending previous work.
problem Proving Quantitative Fatou Theorem on Riemannian manifolds.
method Extending ε-approximation lemma to manifold setting.
result Proves Quantitative Fatou Theorem for Lipschitz domains on Riemannian manifolds.
In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kähler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds c…
New vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.
problem Vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.
method Utilized refined Kato type inequalities and Böchner technique to generalize results to Lp-integrable pluriharmonic functions and harmonic 1-forms. result Proved vanishing property of pluriharmonic functions with finite Lp energy on complete Kähler manifolds. Study on transnormal functions and their level sets on Finsler manifolds.
problem Understanding transnormal functions and their geometric properties on Finsler manifolds.
method Proving smoothness of focal varieties and regular level sets of transnormal functions.
result Focal varieties of a C2 transnormal function are smooth submanifolds and regular level sets are tubes over these varieties.
The paper splits manifolds using infinity harmonic functions with linear growth.
problem Splitting manifolds with specific harmonic functions.
method Analyzes manifolds with non-negative Ricci or sectional curvature, focusing on infinity harmonic functions with linear growth.
result Extends Savin's theorem to surfaces with non-negative sectional curvature.
The study finds a special type of smooth function on connected sums of manifolds.
problem Finding smooth functions that are Morse on preimages of non-extrema values.
method Investigates internally Morse (I-Morse) and neat with respect to Reeb graph (N-Reeb) functions.
result Constructs an IN-Morse-Reeb function on a connected sum of given manifolds.
The questions when two Morse function on closed manifolds are conjugated is investigated. Using the handle decompositions of manifolds the condition of conjugation is formulated. For each Morse function on 3-manifold the ordered generalized Heegaard diagram is built. The criteria of Morse function conjugation are given…
Study sharp decay of capacity for subharmonic functions on compact Hermitian manifolds.
problem Sharp decay of capacity of sublevel sets of (ω,m)-subharmonic functions. method Generalizes previous results on Kähler manifolds and obtains full characterizations of polar sets.
result Full characterizations of polar sets and extremal functions.
Method calculates function integrals on complex manifolds.
problem Integrating functions on complex manifolds.
method Digital representation and calculation method.
result Integral calculation on compact manifolds.
A new method for functional data classification using supervised manifold learning.
problem Improving classification of functional data.
method Supervised manifold learning approach considering label information.
result The proposed method achieves highly competitive performance compared to existing methods.
Estimates for harmonic functions in curved spaces.
problem Quantifying harmonic functions in curved spaces.
method Quantitative Sobolev estimates for p-harmonic functions in manifolds with curvature conditions. result Established a quantitative second order Sobolev estimate for p-harmonic functions. Researchers study surface area functionals in CR manifolds, deducing equations for various cases.
problem Investigating surface area functionals in 3D CR manifolds.
method Deduced Euler-Lagrange equations for energy functionals in various 3D CR manifolds.
result New equations deduced for surface area functionals on disk bundles, Rossi spheres, and 3D tori.
Paper discusses directional differentiability of interval-valued functions on Riemannian manifolds.
problem Equivalence of directional differentiability of interval-valued functions and their components.
method Analyzes directional differentiability of interval-valued functions on Riemannian manifolds.
result Directional differentiability of interval-valued functions is not equivalent to the directional differentiability of their components.
Proves Kastler-Kalau-Walze theorem for spectral Einstein functional on low-dimensional manifolds.
problem Proving Kastler-Kalau-Walze type theorems for spectral Einstein functional.
method Defining spectral Einstein functional associated with Dirac operator and proving theorem for low-dimensional manifolds.
result Proves Kastler-Kalau-Walze type theorem for spectral Einstein functional on low-dimensional manifolds with boundary.
Defines spectral Einstein functionals for sub-Dirac operators on manifolds with boundary.
problem Calculating Einstein-like functionals for sub-Dirac operators.
method Introduced spectral Einstein functional for sub-Dirac operators on manifolds with boundary.
result Proved a theorem for spectral Einstein functions on four-dimensional manifolds.
The study proves exotic smooth structures and equivalent genus functions in 4-manifolds.
problem Understanding exotic smooth structures and genus functions in 4-manifolds.
method Proving the existence of infinitely many exotic structures and equivalence of genus functions.
result Genus functions of exotic 4-manifolds are pairwise equivalent and stable under certain operations.
If the Killing vector field in a Riemannian manifold is the gradient of a smooth real valued function, then it is called Killing potential. In this paper we have deduced a necessary condition for the existence of Killing potential in a complete Riemannian manifold. Yau proved the Liouville theorem of harmonic function …
The paper defines a new functional and proves related theorems for manifolds with boundary.
problem Defining and proving theorems for manifolds with boundary.
method Defining the spectral Einstein functional and relating it to the noncommutative residue.
result Proof of Dabrowski-Sitarz-Zalecki type theorems for spectral Einstein functional on 4D manifolds with boundary.
Gradient estimates for special harmonic functions on manifolds.
problem Estimating gradients of (p,V)-harmonic functions on Riemannian manifolds. method Using Moser iteration method, volume comparison theorem, and Sobolev embedding theorem.
result Explicit global gradient estimates for positive entire (p,V)-harmonic functions. In much of the literature on function approximation by deep networks, the function is assumed to be defined on some known domain, such as a cube or a sphere. In practice, the data might not be dense on these domains, and therefore, the approximation theory results are observed to be too conservative. In manifold learni…
We derive a class of variational functionals which arise naturally in conformal geometry. In the special case when the Riemannian manifold is locally conformal flat, the functional coincides with the well studied functional which is the integration over the manifold of the k-symmetric function of the Schouten tensor of…
Extends Kodaira Spencer to higher-dimensional Calabi-Yau manifolds.
problem No specific problem stated; extension of functional.
method Extension to Calabi-Yau manifolds of arbitrary dimension.
result Extension of the Kodaira Spencer functional.
Constructs explicit p-harmonic functions on Grassmannians and flag manifolds.
problem Finding proper p-harmonic functions on Grassmannians and flag manifolds. method Using the method of eigenfamilies to construct explicit functions.
result Explicit complex-valued proper p-harmonic functions on compact real Grassmannians and non-descending functions on real flag manifolds. The paper introduces spectral Einstein functionals for Dirac operators on manifolds with boundary.
problem Analyzing perturbations of Dirac operators on manifolds with boundary.
method Introduction of spectral Einstein functionals and proof of Dabrowski-Sitarz-Zalecki type theorems.
result Proof of theorems associated with spectral Einstein functionals for perturbations of Dirac operators.
New L-functions for 3-manifolds connect to Witten invariants and relate to generalized Bernoulli polynomials.
problem Understanding L-functions for 3-manifolds and their invariants. method Using Mellin transforms and asymptotic techniques, proving entire functions and their values.
result Linear relations between L-function values at negative integers, generalizing known zeta functions. Kähler information manifolds for signal filters in weighted Hardy spaces are explored.
problem Developing a geometric framework for signal processing filters in weighted Hardy spaces.
method Introducing weighted Hardy spaces and smooth transformations of transfer functions, demonstrating the Kähler manifold structure.
result The Riemannian geometry of weighted Hardy norms for transfer functions forms a Kähler manifold.
Study examines metrics and functionals for Hodge-Dirac operator on manifolds.
problem Examining metrics and functionals for Hodge-Dirac operator on manifolds.
method Analyzing the metric and Einstein bilinear functionals of differential forms for Hodge-Dirac operator d+δ. result The functionals reproduce those for the canonical Dirac operator on a spin manifold up to a numerical factor.