About

Robin Neumayer is an associate professor in the Department of Mathematical Sciences at Carnegie Mellon University. His research lies at the interface of the calculus of variations, partial differential equations (PDE), and geometric analysis. He has contributed to the understanding of isoperimetric and Sobolev-type inequalities, including their stability and rigidity properties, and has worked on problems related to scalar curvature, eigenfunction stability, and geometric measure theory. His work is partly supported by NSF CAREER and NSF RTG grants, and he has a background that includes postdoctoral fellowships at Northwestern University and participation in the Variational Methods in Geometry special year at the Institute for Advanced Study. He completed his Ph.D. at UT Austin under the supervision of Alessio Figalli and Francesco Maggi.

Research topics

• Machine Learning
• Computer Science
• Mathematical analysis
• Combinatorics
• Geometry
• Mathematics
• Algorithm

Selected publications

• Rectifiability and uniqueness of blow-ups for points with positive Alt–Caffarelli–Friedman limit

  Mathematische Annalen · 2025-01-06 · 1 citations

  articleOpen accessSenior authorCorresponding

  Abstract We study the regularity of the interface between the disjoint supports of a pair of nonnegative subharmonic functions. The portion of the interface where the Alt–Caffarelli–Friedman (ACF) monotonicity formula is asymptotically positive forms an $$\mathcal {H}^{n-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> -rectifiable set. Moreover, for $$\mathcal {H}^{n-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> -a.e. such point, the two functions have unique blowups, i.e. their Lipschitz rescalings converge in $$W^{1,2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> to a pair of nondegenerate truncated linear functions whose supports meet at the approximate tangent plane. The main tools used include the Naber–Valtorta framework and our recent result establishing a sharp quantitative remainder term in the ACF monotonicity formula. We also give applications of our results to free boundary problems.

  PublisherOA PDFDOI

• Local minimizers of the anisotropic isoperimetric problem on closed manifolds

  Indiana University Mathematics Journal · 2025-01-01

  articleSenior author
  PublisherDOI

• On the non-uniqueness of locally minimizing clusters via singular cones

  ArXiv.org · 2025-07-18

  preprintOpen access

  We construct partitions of $\mathbb{R}^n$ into three sets $\{\mathscr{X}(1),\mathscr{X}(2),\mathscr{X}(3)\}$ that locally minimize interfacial area among compactly supported volume preserving variations and that blow down at infinity to singular area-minimizing cones. As a consequence, we prove the non-uniqueness of the standard lens cluster in a large number of dimensions starting from $8$.

  PublisherOA PDFDOI

• Quantitative Resolvent and Eigenfunction Stability for the Faber-Krahn Inequality

  ArXiv.org · 2025-04-17

  preprintOpen accessSenior author

  For a bounded open set $Ω\subset \mathbb{R}^n$ with the same volume as the unit ball, the classical Faber-Krahn inequality says that the first Dirichlet eigenvalue $λ_1(Ω)$ of the Laplacian is at least that of the unit ball $B$. We prove that the deficit $λ_1(Ω)- λ_1(B)$ in the Faber-Krahn inequality controls the square of the distance between the resolvent operator $(-Δ_Ω)^{-1}$ for the Dirichlet Laplacian on $Ω$ and the resolvent operator on the nearest unit ball $B(x_Ω)$. The distance is measured by the operator norm from $L^{\infty}$ to $L^2$. As a main application, we show that the Faber-Krahn deficit $λ_1(Ω)- λ_1(B)$ controls the squared $L^2$ norm between $k$th eigenfunctions on $Ω$ and $B(x_Ω)$ for every $k \in \mathbb{N}.$ In both of these main theorems, the quadratic power is optimal.

  PublisherOA PDFDOI

• Rigidity of critical points of hydrophobic capillary functionals

  ArXiv.org · 2025-09-26

  preprintOpen access

  We prove the rigidity, among sets of finite perimeter, of volume-preserving critical points of the capillary energy in the half space, in the case where the prescribed interior contact angle is between $90^\circ$ and $120^\circ$. No structural or regularity assumption is required on the finite perimeter sets. Assuming that the ``tangential'' part of the capillary boundary is $\mathcal{H}^n$-null, this rigidity theorem extends to the full hydrophobic regime of interior contact angles between $90^\circ$ and $180^\circ$. Furthermore, we establish the anisotropic counterpart of this theorem under the assumption of lower density bounds.

  PublisherOA PDFDOI

• Quantitative estimates for the relative isoperimetric problem and its gradient flow outside convex bodies in the plane

  ArXiv.org · 2025-08-28

  preprintOpen access

  We prove three related quantitative results for the relative isoperimetric problem outside a convex body $Ω$ in the plane: (1) Łojasiewicz estimates and quantitative rigidity for critical points, (2) rates of convergence for the gradient flow, and (3) quantitative stability for minimizers. These results come with explicit constants and optimal exponents/rates, and hold whenever a simple two-dimensional auxiliary variational problem for circular arcs outside of $Ω$ is nondegenerate. The proofs are inter-related, and in particular, for the first time in the context of isoperimetric problems, a flow approach is used to prove quantitative stability for minimizers.

  PublisherOA PDFDOI

• dp–convergence and 𝜖–regularity theorems forentropy and scalar curvature lower bounds

  Geometry & Topology · 2023-05-01 · 11 citations

  articleOpen accessSenior author

  Consider a sequence of Riemannian manifolds .M n i ; g i / whose scalar curvatures and entropies are bounded from below by small constants R i ; i i .The goal of this paper is to understand notions of convergence and the structure of limits for such spaces.As a first issue, even in the seemingly rigid case i !0, we will construct examples showing that from the Gromov-Hausdorff or intrinsic flat points of view, such a sequence may converge wildly, in particular to metric spaces with varying dimensions and topologies and at best a Finsler-type structure.On the other hand, we will see that these classical notions of convergence are the incorrect ones to consider.Indeed, even a metric space is the wrong underlying category to be working on.Instead, we will introduce a weaker notion of convergence called d p -convergence, which is valid for a class of rectifiable Riemannian spaces.These rectifiable spaces will have a well-behaved topology, measure theory and analysis.This includes the existence of gradients of functions and absolutely continuous curves, though potentially there will be no reasonably associated distance function.Under this d p notion of closeness, a space with almost nonnegative scalar curvature and small entropy bounds must in fact always be close to Euclidean space, and this will constitute our -regularity theorem.In particular, any sequence .M n i ; g i / with lower scalar curvature and entropies tending to zero must d p -converge to Euclidean space.More generally, we have a compactness theorem saying that sequences of Riemannian manifolds .M n i ; g i / with small lower scalar curvature and entropy bounds R i ; i must d p -converge to such a rectifiable Riemannian space X .In the context of the examples from the first paragraph, it may be that the distance functions of M i are degenerating, even though in a well-defined sense the analysis cannot be.Applications for manifolds with small scalar and entropy lower bounds include an L 1 -Sobolev embedding and a priori L p scalar curvature bounds for p < 1. 53C21

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• Convergence and Regularity of Manifolds with Scalar Curvature and Entropy Lower Bounds

  WORLD SCIENTIFIC eBooks · 2023-01-08

  book-chapterSenior author
  PublisherDOI

• Rigidity theorems for best Sobolev inequalities

  Advances in Mathematics · 2023-10-17

  articleOpen accessCorresponding

  For n≥2, p∈(1,n), the “best p-Sobolev inequality” on an open set Ω⊂Rn is identified with a family ΦΩ of variational problems with critical volume and trace constraints. When Ω is bounded we prove: (i) for every n and p, the existence of generalized minimizers that have at most one boundary concentration point, and: (ii) for n>2p, the existence of (classical) minimizers. We then establish rigidity results for the comparison theorem “balls have the worst best Sobolev inequalities” by the first named author and Villani, thus giving the first affirmative answers to a question raised in [14].

  PublisherDOI

• Local Minimizers of the Anisotropic Isoperimetric Problem on Closed Manifolds

  arXiv (Cornell University) · 2023-08-08

  preprintOpen accessSenior author

  Local minimizers for the anisotropic isoperimetric problem in the small-volume regime on closed Riemannian manifolds are shown to be geodesically convex and small smooth perturbations of tangent Wulff shapes, quantitatively in terms of the volume.

  PublisherOA PDFDOI

Recent grants

• Stability of Functional and Geometric Inequalities and Applications

  NSF · $83k · 2019–2021

Frequent coauthors

• Dennis Kriventsov

  6 shared

• Francesco Maggi

  The University of Texas at Austin

  6 shared

• M. Allen

  Observatoire astronomique de Strasbourg

  3 shared

• Mark G. Allen

  3 shared

• Marco Caroccia

  3 shared

• Man-Chun Lee

  3 shared

• Rustum Choksi

  McGill University

  3 shared

• Aaron Naber

  Northwestern University

  3 shared

Education

• Ph.D.

  University of Texas, Austin