We present a black hole effect by strategically leveraging a conformal
    mapping in elastic continuum with curved-space framework, which is less
    stringent compared to a Schwarzschild model transformed to isotropic refractive
    index profiles. In the conformal map approach, the 2D point singularity
    associated to the black hole effect is accomplished by physical plates with
    near-to-zero thickness. The analog gravity around the singularity results in
    highly confined energy and lagged timings within a branch cut of the conformal
    map. These effects are quantified both numerically and experimentally in
    reference to control trials in which the thickness is not modulated. The
    findings would deepen our understanding of the elastic analog in mimicking
    gravitational phenomena, as well as establish the elastic continuum framework
    for developing a generic design recipe in the presence of the index
    singularity. Geometric landscapes with elastically curved surfaces would be
    applicable in a variety of applications such as sensing, imaging, vibration
    isolation, and energy harvesting.

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