Fixed-concavity collar collapse and failure of the raw double

Definitions and mathematical statement

The source asks to study collapse of Riemannian manifolds with boundary, with $\sec\geq K$ in the interior and controlled boundary concavity. It does not specify dimension, compactness, diameter, concavity convention or a desired structural theorem.

The aid gives an explicit compact model. Let $(N^m,h)$ be a closed connected Riemannian manifold with $m\geq1$ and $\sec_h\geq0$, and fix $\lambda\geq0$. For $\varepsilon>0$ put \[ M_\varepsilon=N\times[0,\varepsilon],\qquad g_\varepsilon={\,\mathrm{d}} t^2+a_\varepsilon(t)^2h,\qquad a_\varepsilon(t)=1+\frac\lambda\varepsilon t(\varepsilon-t). \] Use the inward unit normal and the convention $\mathrm{II}(X,Y)=-g(\nabla_X\nu,Y)$, so convex Euclidean boundaries have positive form. Since $a(0)=a(\varepsilon)=1$, $a'(0)=\lambda$ and $a'(\varepsilon)=-\lambda$, both boundary faces satisfy \[ \mathrm{II}=-\lambda\,g_{\partial M_\varepsilon}. \] Thus concavity is fixed rather than deteriorating with collapse.

The warped-product sectional curvatures are \[ \sec(\partial_t,X)=-\frac{a''}{a}=\frac{2\lambda}{\varepsilon a}\geq0, \qquad \sec(X,Y)=\frac{\sec_h(X,Y)-(a')^2}{a^2}\geq-\lambda^2 \] for the corresponding mixed and tangential planes; the second formula is used when such tangential planes exist. The aid reports the exact inradius and quantitative collapse \[ \operatorname{inrad}(M_\varepsilon)=\varepsilon/2,\qquad d_{\mathrm{GH}}(M_\varepsilon,N)\leq\varepsilon/2, \] and volume \[ \operatorname{Vol}(M_\varepsilon)=\operatorname{Vol}(N)\int_0^\varepsilon a_\varepsilon(t)^m\,{\,\mathrm{d}} t \sim\varepsilon\operatorname{Vol}(N). \] These are the intrinsic length metrics; projection to $N$ supplies the stated correspondence.

Let $D M_\varepsilon$ be the raw metric double, gluing two copies by the identity on both boundary faces, without smoothing. The aid asserts \[ d_{\mathrm{GH}}(D M_\varepsilon,N)\leq\varepsilon/2, \] but for $\lambda>0$ each individual double has no finite Alexandrov lower curvature bound. It attributes this to Kosovskii's necessary gluing condition: the sum of the two face forms is $-2\lambda g$, which is negative in every nonzero tangent direction. Positive base dimension is therefore retained in this obstruction. At $\lambda=0$ the product double has no such negative-form obstruction.

The model shows that controlled concavity and an interior lower curvature bound do not justify applying ordinary lower-curvature gluing directly to raw doubles, even when both original spaces and doubles collapse to a smooth nonnegatively curved base. It does not classify all boundary collapse or exclude other smoothing, extension or structural approaches to the original question.

Short English statement

The supplied collar family collapses with fixed boundary concavity and a uniform interior curvature lower bound. Its raw doubles still have no finite Alexandrov lower bound when the concavity is negative, so direct doubling is not automatically a valid reduction.

Sources

• Local UnsolvedMath registry, permanent ID $20003145$, source code catalogue entry; canonical aim-topology-notes.json, notes[232].
• AIM workshop Manifolds with nonnegative sectional curvature, Problem 20: https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf. Original remarks and literature field are empty.
• The aid invokes Kosovskii's necessary gluing condition without an authoritative full citation or full theorem hypotheses.
• Research attempt 1: partial_result, status partially_solved, classification PARTIAL-PROGRESS, rigor proved, novelty confidence low. Its explicit collar and raw-double obstruction are machine-generated aid, not peer reviewed, and require independent verification before citation.