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Abstract

Let Ω⊂ℝ^N, N≥2, be a bounded connected domain with smooth boundary. Consider the Dirichlet heat evolution with initial temperature identically one. We prove that if its signed outward normal derivative is spatially constant on the entire boundary at one positive time, then Ω is a ball. No connectedness assumption is imposed on the boundary. The argument uses the classical method of moving planes, with an L^2 comparison at the incompatible initial corner, a component reconstruction lemma, and an explicit parabolic corner barrier. In particular, the result gives a negative answer in the smooth connected Euclidean class to the finite-observation nonball scenario in Cavallina--Pinamonti's Problem 3 and Conjecture 7.4. We also prove a local quantitative estimate near a ball: at any one time in a fixed compact positive-time interval, the full boundary flux defect in C^(m-1,α) controls a volume- and centroid-normalized radial-graph deformation in C^(m,α), for m≥3. On mean-zero deformations, the linearized flux has precisely the translation kernel and strictly positive degree-ℓ≥2 spherical multipliers with order-one growth. This quantitative result is local, uses strong boundary norms, and has no additive short-time error term. For each fixed positive time, restricting the observation to a moving proper boundary cap instead gives an infinite-dimensional family of nearby normalized nonballs with exactly the ball flux on that cap. The initial datum remains essential.