Concave Barrier Functions in Riemannian Geometry

• Concave barrier functions are defined via Busemann functions on geodesic lines in complete Riemannian manifolds, ensuring a quantitative obstruction near boundary points.
• They exhibit local semi-concavity with linear modulus, leading to almost everywhere twice differentiability and robust control over singular structures.
• These functions are pivotal in classifying geodesics, organizing manifold foliations, and linking geometric analysis with dynamical systems and Hamilton–Jacobi theory.

A concave barrier function is a mathematical construct that “enforces” an obstruction by growing rapidly—often diverging—as its argument approaches the boundary of a designated set. Such functions are widely used in Riemannian geometry, optimization, control theory, statistical inference, and nonsmooth analysis, typically to ensure safety, regularity, or tractable control over dynamical or geometric processes. In the context of non-compact, complete, boundaryless, connected Riemannian manifolds, concave barrier functions emerge naturally as sums of Busemann functions associated with lines (geodesics) in the manifold. These functions adopt local and sometimes global concavity properties, which enable rigorous classification, regularity, and structure in geometric and dynamical settings.

1. Construction via Busemann Functions and Lines

Given a complete, non-compact, boundaryless, connected Riemannian manifold (M, g), a line is defined as a geodesic $y: \mathbb{R} \to M$ such that $d(y(t_1), y(t_2)) = |t_2 - t_1|$ for any $t_1, t_2 \in \mathbb{R}$. Associated to each line are two rays:

• $y^+ \equiv y|_{[0, +\infty)}$ (“forward” ray)
• $y^- \equiv y|_{(-\infty, 0]}$ (“backward” ray)

The Busemann function $b_y$ corresponding to a given ray $y$ is defined by

$b_y(x) = \lim_{t \to +\infty} [d(x, y(t)) - t]$

for $x \in M$. For each line $y$, let $b_y^+$ and $b_y^-$ denote the Busemann functions for $y^+$ and $y^-$, respectively.

Barrier Function:

The barrier function associated to $y$ is then

$B_y(x) = b_y^+(x) + b_y^-(x)$

which serves as a quantitative obstruction to approaching both asymptotic directions of the line. The construction is deeply tied to the geodesic structure of the manifold, and $B_y$ acts as a “barrier” in both forward and backward senses.

2. Regularity and Semi-Concavity

A principal result is that each Busemann function $b_y^{\pm}$ is a viscosity solution of the Hamilton–Jacobi equation defined by the metric:

$|du|^2 = 1$

Consequently, by the main theorem, $b_y^{\pm}$ are locally semi-concave with linear modulus. Explicitly, for any open set $\Omega \subset \mathbb{R}^n$ and relatively compact convex subset $Q \subset \Omega$, there exists $C > 0$ so that

$u(x) - \frac{C}{2}|x|^2$

is concave on $Q$. Since the sum of two locally semi-concave functions (with linear modulus) is also locally semi-concave with the same modulus, the barrier function $B_y$ inherits this property. By Alexandroff’s theorem, $B_y$ is almost everywhere twice differentiable.

Function type	Concavity property	Regularity
Busemann	Locally semi-concave	Viscosity solution, locally Lipschitz
Barrier ($B_y$)	Semi-concave with linear modulus	Twice differentiable a.e.

This semi-concavity ensures powerful control over singularity structures and invariance properties, facilitating analysis and computations.

3. Invariance, Nonnegativity, and Boundary Behavior

Barrier functions constructed in this way enjoy two critical invariance properties:

• Time translation invariance: For $y_T(t) = y(t + T)$, $B_{y_T}(x) = B_y(x)$ for all $T \in \mathbb{R}$.
• Nonnegativity: $B_y(x) \geq 0$ for all $x \in M$, with $B_y(x) = 0$ precisely on a distinguished set $G_y$.

For $x \in G_y$, there exists a unique line through $x$ such that its forward and backward rays are corays of $y^+$ and $y^-$, respectively. The structure of $G_y$ leads to a lamination or foliation of the manifold by lines aligned with $y$.

Additionally, on any compact subset $K \subset M$, there exists $C > 0$ so that

$B_y(x) \leq C \cdot d^2(x, y)$

which is a sharp quadratic control property common to semi-concave barrier functions.

4. Concavity and Geometric Implications

In classical settings with nonnegative sectional curvature, each Busemann function is concave along geodesics, so their sum $B_y$ is strictly concave—making $B_y$ a genuine concave barrier function. However, the general formulation admits only local semi-concavity unless further geometric restrictions are imposed. This generalized notion is analogously related to Mather’s barrier functions in weak KAM theory, enabling classification and geometric analysis even without global curvature assumptions.

The interpretation of $B_y$ as a concave barrier arises naturally in dynamical systems and Riemannian rigidity phenomena, especially as a tool to classify lines and to define relations (e.g., an order $<$ and equivalence $\sim$) between them, based on their asymptotic boundary behavior.

5. Role in the Hamilton–Jacobi Equation and Viscosity Solutions

Since each $b_y^{\pm}$ is a global viscosity solution of $|du|^2 = 1$, the barrier function $B_y$ inherits key stability and regularity properties of such solutions under operational limits. At differentiable points,

$|\nabla u(x)|^2 = 1$

holds for each Busemann function, and thus

$B_y(x)$

links the geometric flows on $M$ to analytic properties governed by Hamilton–Jacobi theory. These connections bring methods from nonsmooth analysis and weak KAM theory into Riemannian geometry, providing tools for studying singularities, boundaries at infinity (ideal boundaries), and global geometric structure.

6. Applications: Classification, Foliation, and Boundary Structure

Barrier functions are pivotal in the following areas:

• Classification of lines: Relations derived from the barrier function ($<$, $\sim$) organize the lines in $M$, grouping lines that “connect” the same pair of ideal boundary elements.
• Lamination/foliation: The set $G_y$ (where $B_y(x) = 0$) is foliated by lines functionally determined by the barrier function, enabling a geometric decomposition of the manifold.
• Connections to ideal boundaries: Through barrier and Busemann functions, one can define an ideal boundary of $M$, related to geodesic rays and horofunctions, giving mathematical structure to “points at infinity” and facilitating classification of infinite geodesic behavior.
• Rigidity and dynamical systems: Barrier functions are robustly motivated by Lagrangian dynamical systems and Mather theory, useful for connecting orbits and analyzing rigidity conjectures involving geodesic uniqueness and completeness.

7. Summary of Key Formulas

• Busemann function for a ray $y$:

$$b_y(x) = \lim_{t \rightarrow +\infty} [d(x, y(t)) - t]$$

• Barrier function for a line $y$:

$B_y(x) = b_y^+(x) + b_y^-(x)$

• Viscosity solution of the Hamilton–Jacobi equation:

$|du(x)|^2 = 1$

• Local semi‐concavity with linear modulus:

$$u(x) - \frac{C}{2}|x|^2 \quad \text{is concave (locally)}$$

Conclusion

Concave barrier functions on Riemannian manifolds, as constructed from Busemann functions of geodesic lines, form a mathematically rigorous class of objects—locally semi-concave with linear modulus, almost everywhere twice differentiable, and intimately connected to the Hamilton–Jacobi equation and weak KAM theory. Their invariance properties, connections to ideal boundary constructions, classification of geodesics, and analytic regularity underpin their utility in Riemannian geometry, dynamical systems, and geometric analysis. In curvature-constrained settings, these barrier functions are strictly concave, but in more general situations they retain all the desirable local properties necessary for advanced mathematical and analytic applications.