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Almost prescribing scalar curvature by mixed convex integration

Fatine Aliouane, Ludovic Rifford, Mélanie Theillière

TL;DR

The paper develops a mixed convex integration framework that couples first- and second-order corrugations to solve a class of semilinear second-order PDEs, enabling a constructive route to manipulate scalar curvature. By applying this method to perturbations on tori, it provides a new, explicit proof that one can produce smooth metrics with scalar curvature closely controlled relative to a given background metric, recovering Lohkamp-type results in dimensions $n\ge 3$. The approach proceeds through a torus-based analysis (flat and general metrics) and culminates with a triangulation-based global argument on manifolds, using a thickening technique to achieve localized scalar-curvature deficits. The work offers a self-contained, geometric-analytic construction that yields almost-prescribed scalar curvature metrics with precise $C^0$-closeness and sharp inequalities, contributing a flexible alternative to Lohkamp's original perturbative strategy and broadening the toolbox for curvature control in Riemannian geometry.

Abstract

We introduce a method of mixed convex integration and demonstrate its suitability for solving a particular class of semilinear second-order partial differential relations. As an application, we provide a new proof of a result on scalar curvature originally established by Lohkamp.

Almost prescribing scalar curvature by mixed convex integration

TL;DR

The paper develops a mixed convex integration framework that couples first- and second-order corrugations to solve a class of semilinear second-order PDEs, enabling a constructive route to manipulate scalar curvature. By applying this method to perturbations on tori, it provides a new, explicit proof that one can produce smooth metrics with scalar curvature closely controlled relative to a given background metric, recovering Lohkamp-type results in dimensions . The approach proceeds through a torus-based analysis (flat and general metrics) and culminates with a triangulation-based global argument on manifolds, using a thickening technique to achieve localized scalar-curvature deficits. The work offers a self-contained, geometric-analytic construction that yields almost-prescribed scalar curvature metrics with precise -closeness and sharp inequalities, contributing a flexible alternative to Lohkamp's original perturbative strategy and broadening the toolbox for curvature control in Riemannian geometry.

Abstract

We introduce a method of mixed convex integration and demonstrate its suitability for solving a particular class of semilinear second-order partial differential relations. As an application, we provide a new proof of a result on scalar curvature originally established by Lohkamp.
Paper Structure (18 sections, 13 theorems, 201 equations, 4 figures)

This paper contains 18 sections, 13 theorems, 201 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mixed convex integration
  3. A few notations
  4. Mixed corrugation processes
  5. Semilinear second-order partial differential relations
  6. Geometrical interpretation of Proposition \ref{['PROPmixedapp']}
  7. Almost prescribing scalar curvature on tori
  8. Perturbations of the flat metric
  9. Perturbations of a general metric on $\mathbb T^n$
  10. Prescribing scalar curvature on thick tori
  11. Proof of Theorem \ref{['THMScalar']}
  12. Ricci and scalar curvatures in Riemannian geometry
  13. Main definitions
  14. Ricci and scalar curvatures in the general case
  15. Ricci and scalar curvature of diagonal metrics
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $(M,g)$ be a smooth Riemannian manifold of dimension $n\geq 3$, then for every smooth Riemannian metric $g_0$ on $M$, every smooth function $k: M \rightarrow (0,+\infty)$, and every $\epsilon >0$, there exists a smooth Riemannian metric $g_{\epsilon}$ on $M$ satisfying

Figures (4)

  • Figure 1: The surface $2(w_1+w_2+v_1^2)=F$, where $F$ is a constant
  • Figure 2: The codimension $2$ manifold $\tilde{\Sigma}^3$ associated with $D^3$, shown in red, is the union of a vertical segment containing the origin and a planar circle centered at the origin
  • Figure 3: The union of $U$ (in blue) and $D$ (in yellow) covers the simplex $\varphi_{\mathcal{K}}(\sigma)$. The green region corresponds to points belonging to both $U$ and $D$
  • Figure 4: The sets $U'$, $\Sigma^2$, and $T$ in the case where $\varphi_{\mathcal{K}}(\sigma)$ is a $2$-dimensional simplex in ambient dimension $3$

Theorems & Definitions (25)

  • Theorem 1.1
  • Definition 2.1: Integral-loop operator
  • Proposition 2.2
  • proof : Proof of Proposition \ref{['PROPmixed']}
  • Definition 2.3
  • Proposition 2.4
  • proof : Proof of Proposition \ref{['PROPmixedapp']}
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['PROPTnFlat']}
  • Remark 3.2
  • ...and 15 more