Open research questions in Geometric Analysis and Curvature Flows
230 gap statements mined from Geometric Analysis and Curvature Flows papers in our 4.5M-paper local library, which holds 1,318 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.
What the literature leaves open
The gap is in the understanding of curvature and connections on these manifolds.
Riemannian Geometry in Multiplicative Analysis: Curvature, Connections, and Isomorphic Structures · 2026 · DOIThe paper does not provide a comprehensive analysis of all possible graph properties. The authors focus on the relationship between doubling graphs and the Cheeger isoperimetric inequality, leaving other properties unexplored.
Future research may focus on extending the results to more general cases. It could explore the applications of the new techniques to other areas of spin geometry and monopole Floer homology.
It highlights the need for new techniques to study the locus of operators with non-trivial kernel.
The study of timelike conformal vector fields on Lorentzian manifolds has been widely studied, but the 3-dimensional case has attracted substantial attention in the literature. There is a need to understand the geometry of the flow generated by these fields.
This paper provides a new viewpoint on the geometric analysis of soliton surfaces produced from the B-DR equation and establishes the foundation for future studies in this area.
Geometric Analysis of Soliton Surfaces via Generalized Bishop Frame Field of Type-C Associated with Betchov–Da Rios Equation in \({\mathbb{E}^{4}}\) · 2026 · DOIThe lack of heat kernel estimates for the Ricci-harmonic flow. The need to generalize the previous work of Bamler–Zhang on Ricci flow to the Ricci-harmonic flow setting.
Future research can focus on applying the stability estimates obtained in the paper to problems in differential geometry and geometric analysis. The authors suggest studying the behavior of nearly spherical sets in space forms using the stability estimates. Future research can also explore generalizing the results to other types of geometric inequalities.
There is a gap in the existing literature regarding the stability estimates for weighted Alexandrov-Fenchel type inequalities. The paper identifies a need for generalizing earlier results to broader geometric settings and convex weight functions.
The stability of the cut locus under perturbations of the metric was an open question. The convergence of Riemannian metrics was not well understood. The properties of the cut locus were not fully understood.
Future research can explore the applications of the location-scale-shape model. Future research can investigate the properties of the Wasserstein geometry of the model.
The study addresses the need for a new class of statistical models that can generalize several distributions used in extreme-value theory.
The paper does not provide a general solution for all possible chains of Lie subgroups. The results are limited to the case of compact Lie groups and homogeneous spaces.
Future research directions include the study of sub-Riemannian geodesics on non-compact Lie groups. The authors suggest studying the geometry of sub-Riemannian manifolds using the framework provided in the paper.
Previous results were limited to a restricted range of curvature values. The existence of smooth complete hypersurfaces in hyperbolic space with constant curvature and a prescribed asymptotic boundary at infinity was not established for all possible curvature values.
Future research could involve studying other types of connections in semi-Riemannian manifolds. Future research could involve generalizing the results to other types of manifolds.
The gap is the lack of understanding of how to induce statistical connections in null hypersurfaces. The gap is the lack of understanding of the properties of statistical connections in indefinite Kähler manifolds.
The authors highlight the need for a coherent analytic and geometric framework for studying these surfaces.
Möbius-type minimal surfaces family in R 4 ${\mathbb{R}}^{4}$ via the generalized Weierstrass–Enneper representation · 2026 · DOIThe lack of characterization of L-isotropic hypersurfaces in R n+1. The need for a rigidity theorem for L-isotropic hypersurfaces parameterized by lines of curvature.
There is a need to better understand the behavior of ancient solutions of mean curvature flow. There is a need to extend planarity and convexity estimates to higher codimension.
The paper establishes that in the odd first Betti number case, the diffeomorphism φ defining the Kähler mapping torus is either isotopic to the identity or of finite order, leading to one of three specific Thurston geometries (S² × S¹, H² × ℝ/Γ, or T²-bundles), but does not provide criteria for determining which Thurston geometry is realized given a specific timelike conformal vector field.
The paper suggests studying the sharpness of the constant c n. The paper suggests exploring modified pinching conditions. The paper suggests investigating the behavior of ancient solutions in low dimensions.
Further analysis of the relationship between doubling graphs and other graph properties. Exploration of the implications of the paper's findings for various fields, such as computer science and mathematics.
Further study of the properties of weakly cyclic Schouten-symmetric manifolds. Analysis of the applications of such manifolds. Examination of the relationships between weakly cyclic Schouten-symmetric manifolds and other geometric structures.
Further study of the stability of the cut locus under other types of perturbations. Analysis of the convergence of Riemannian metrics in other contexts. Application of the results to fields such as physics and engineering.
Most-cited papers in Geometric Analysis and Curvature Flows
- Critical Points and Curvature for Embedded Polyhedral Surfaces · American Mathematical Monthly · 1970 · 243 citations
- The Geometric Mean, Matrices, Metrics, and More · American Mathematical Monthly · 2001 · 217 citations
- When Does the Position Vector of a Space Curve Always Lie in Its Rectifying Plane? · American Mathematical Monthly · 2003 · 147 citations
- Helmholtz’s theorem revisited · American Journal of Physics · 1986 · 142 citations
- Higher Curvatures of Curves in Euclidean Space · American Mathematical Monthly · 1966 · 105 citations
- On the Diffeomorphisms of Euclidean Space · American Mathematical Monthly · 1972 · 97 citations
- A Derivation of n-Dimensional Spherical Coordinates · American Mathematical Monthly · 1960 · 92 citations
- On Deciding Whether a Surface is Parabolic or Hyperbolic · American Mathematical Monthly · 1977 · 58 citations
- Magnetic curves in Sasakian manifolds · Journal of Nonlinear Mathematical Physics · 2021 · 55 citations
- The Uniformization Theorem · American Mathematical Monthly · 1981 · 54 citations
Most recent work
- Stable minimal hypersurfaces in $\mathbf{R}^5$ · Annals of Mathematics · 2026
- Energy identity and no neck property for $$\varepsilon $$-harmonic and $$\alpha $$-harmonic maps into homogeneous target manifolds · Calculus of Variations and Partial Differential Equations · 2026
- Generalized Minkowski formulas and rigidity results for anisotropic capillary hypersurfaces · Journal of Mathematical Analysis and Applications · 2026
- General Casorati inequalities and implications for Riemannian maps and Riemannian submersions · Journal of Mathematical Analysis and Applications · 2026
- The weighted ambient metric for manifolds with density · Advances in Mathematics · 2026
- Loop spaces of 𝑛-dimensional Poincaré duality complexes whose (𝑛-1)-skeleton is a co-𝐻-space · Transactions of the American Mathematical Society · 2026
- Positive curvature conditions on contractible manifolds · Mathematische Annalen · 2026
- A splitting theorem for manifolds with nonnegative spectral Ricci curvature and mean convex boundary · Journal of Functional Analysis · 2026
- Area minimizing hypersurfaces modulo p: a geometric free-boundary problem · Journal of Functional Analysis · 2026
- Contractive transport maps from S 2 to nearly spherical surfaces with positive Ricci curvature · Nonlinear Analysis · 2026
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