Open research questions in Geometric Analysis and Curvature Flows

What the literature leaves open

• The paper identifies a gap in the understanding of minimal surfaces in ℝ⁴. 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 · DOI

• The 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.

  On Laguerre Isotropic Hypersurfaces · 2026 · DOI

• 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.

  Planarity and convexity for pinched ancient solutions of mean curvature flow · 2026 · DOI

• 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.

  Planarity and convexity for pinched ancient solutions of mean curvature flow · 2026 · DOI

• The paper suggests that future research should focus on exploring the properties of non-vacuum solutions. The study recommends that future work should investigate the applications of the results to the study of black holes and other compact objects. The paper proposes that future research should aim to develop new theories of gravity that incorporate the results of the study.

  Non-vacuum Metrics for the Newman-Unti-Tamburino Background: A Coordinate-Free Approach to Diverging and Twisting Solutions · 2026 · DOI

• The study identifies a gap in our understanding of non-vacuum metrics for the Newman-Unti-Tamburino background. The paper highlights the need for a coordinate-free approach to diverging and twisting solutions. The study notes that previous work has focused on vacuum solutions, leaving a gap in our understanding of non-vacuum solutions.

  Non-vacuum Metrics for the Newman-Unti-Tamburino Background: A Coordinate-Free Approach to Diverging and Twisting Solutions · 2026 · DOI

• The inherent degeneracy of the metric and the non-integrability of the underlying distribution. The lack of a well-developed theory of heat kernels in the sub-Riemannian setting. The need to adapt techniques from geometric control theory, analysis on Lie groups, and microlocal analysis to the specific challenges of contact structures and Reeb flows.

  Sharp Heat Kernel Estimates and the Spectral Geometry of Reeb Flows on Contact Sub-Riemannian Manifolds · 2026 · DOI

• The complexity of the Helfrich functional, - The difficulty of analyzing the morphology of biological membranes, - The need to consider the effects of external forces on the membrane

  Hyperbolic Geometry and the Helfrich Functional · 2026 · DOI

• The paper must connect the Jacobi operator to the geometry and topology of CMC hypersurfaces. The paper must derive new rigidity results for the area of CMC hypersurfaces. The paper must fill a gap in the literature regarding the connection between the Jacobi operator and geometry.

  First Eigenvalue of Jacobi Operator and Rigidity Results for Constant Mean Curvature Hypersurfaces · 2026 · DOI

• The authors face the challenge of handling weights vanishing at several points. The paper must address the lack of explicit constants in previous results.

  New weighted inequalities on two–manifolds · 2026 · DOI

• Future research could study the Dirichlet problem for other types of equations on Riemannian manifolds. The results could be extended to more general geometric settings.

  The Dirichlet Problem for Monge-Ampère Type Equations on Riemannian Manifolds · 2026 · DOI

• The Dirichlet problem for Monge-Ampère type equations on Riemannian manifolds was not previously studied. Prior work only considered the Monge-Ampère equation in Euclidean space.

  The Dirichlet Problem for Monge-Ampère Type Equations on Riemannian Manifolds · 2026 · DOI

• The existing result does not provide an explicit estimate of the hyperbolicity constant. The quantitative version of the guessing geodesics criterion is missing.

  A Quantitative Guessing Geodesics Theorem · 2026 · DOI

• The paper identifies a gap in the understanding of Markowitz's pseudodistance and its applications. The paper identifies a need for a review of the fundamental properties of the pseudodistance.

  On Markowitz’s pseudodistance for conformal manifolds · 2026 · DOI

• The need for rigidity results under certain conditions for static manifolds with boundary. The gap in prior work is the lack of understanding of the geometric properties of such manifolds.

  Some rigidity results for static three-manifolds with boundary and positive scalar curvature · 2026 · DOI

• To solve the stable Bernstein problem for n ≥ 7. To study the geometry of minimal hypersurfaces in higher dimensions. To study the stability of minimal hypersurfaces.

  Stable Anisotropic Minimal Hypersurfaces in $$\mathbb {R}^{5}$$ and $$\mathbb {R}^{6}$$ · 2026 · DOI

• The paper assumes that the curve γ is non-degenerate. The results are limited to Borel sets in ℝ3. The paper does not provide a complete characterization of the properties of restricted projections.

  On restricted projections to planes in ℝ3 · 2026 · DOI

• To extend the results to more general settings. To provide a complete characterization of the properties of restricted projections. To apply the results to problems in computer science, physics, and engineering.

  On restricted projections to planes in ℝ3 · 2026 · DOI

• The lack of a characterization of Bonnet mates in geometries other than R3. The need to determine when an isometric immersion can be continuously deformed through isometric immersions that preserve the principal curvatures.

  Uniqueness of the Bonnet problem in Thurston geometries · 2026 · DOI

• The paper does not provide a general method for computing the self-linking number of a curve. The results are limited to curves of constant curvature. The proofs are based on convex integration techniques, which may not be applicable to other types of curves.

  Deformations of Curves with Constant Curvature · 2026 · DOI

• To develop a general method for computing the self-linking number of a curve. To extend the results to curves with non-constant curvature. To apply the results to other areas of mathematics and computer science.

  Deformations of Curves with Constant Curvature · 2026 · DOI

• The gap is the lack of understanding of the behavior of SCSF in higher codimension. The gap is the lack of proof of a variant of Huisken's distance comparison principle for reflection symmetric immersed Curve Shortening flow.

  Huisken’s distance comparison principle in higher codimension · 2026 · DOI

• The curvature conditions may not apply in general. The analysis is limited to specific cases, such as the Schwarzschild-Tangherlini black hole and the Reissner-Nordstrom black hole.

  Curvature conditions for generalized singularity theorems · 2026 · DOI

• Future research could further generalize the results to other types of compact trapped submanifolds. The authors' findings may be relevant to the study of other areas, such as cosmological singularity theorems.

  Curvature conditions for generalized singularity theorems · 2026 · DOI

• The lack of closed expressions for the exponential map, the logarithmic map, and the intrinsic distance in warped Segre-Veronese manifolds. The fact that Segre-Veronese manifolds are not geodesically connected in the Euclidean geometry.

  Warped Geometries of Segre–Veronese Manifolds · 2026 · DOI

Most-cited papers in Geometric Analysis and Curvature Flows

• *-conformal η-Ricci solitons in ϵ-Kenmotsu manifolds · Publications de l Institut Mathematique · 2020 · 5 citations
• Lower Bound Eigenvalue Problems of the Compact Riemannian Spin-Submanifold Dirac Operator · Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi · 2020 · 5 citations
• Closed surfaces with different shapes that are indistinguishable by the SRNF · Archivum Mathematicum · 2020 · 4 citations
• Smarandache Curves of Spacelike Anti-Salkowski Curve with a Timelike Principal Normal According to Frenet Frame · Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi · 2020 · 4 citations
• Why Curves Curve: The Geodesics on the Torus · Mathematics Magazine · 2022 · 3 citations
• Curvature and the equivalence problem in sub-Riemannian geometry · Archivum Mathematicum · 2022 · 2 citations
• On the position vector of surface curves in the Euclidean space · Publications de l Institut Mathematique · 2022 · 2 citations
• Certain almost Kenmotsu metrics satisfying the vacuum static equation · Publications de l Institut Mathematique · 2023 · 2 citations
• Characterizations of The Ruled Surfaces due to Modified Frame · Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi · 2022 · 2 citations
• Flatness of location-scale-shape models under the Wasserstein metric · Information Geometry · 2026 · 1 citations

Most recent work

• Flatness of location-scale-shape models under the Wasserstein metric · Information Geometry · 2026
• Foundations of Carrollian geometry · Physics Reports · 2026
• Almost Ricci solitons on pseudo Ricci symmetric spacetime · International Journal of Geometric Methods in Modern Physics · 2026
• Riemannian Geometry in Multiplicative Analysis: Curvature, Connections, and Isomorphic Structures · Mediterranean Journal of Mathematics · 2026
• The Concavity of p-Rényi Entropy Power and Hamilton Type Gradient Estimate for Doubly Nonlinear Diffusion Equation on Riemannian Manifolds · Results in Mathematics · 2026
• Riemannian Geometry of the Free Step-Two Carnot Group · Chinese Annals of Mathematics, Series B · 2026
• Isoperimetric Inequality, p-parabolicity and Doubling Graphs · Results in Mathematics · 2026
• Gradient Schouten Solitons with Respect to the Sasaki Metric · Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics · 2026
• TOPOLOGY OF THE DIRAC EQUATION ON SPECTRALLY LARGE THREE-MANIFOLDS · Journal of the Institute of Mathematics of Jussieu · 2026
• Timelike Conformal Fields on Closed 3-Manifolds · Mediterranean Journal of Mathematics · 2026

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