FIG. 4: The RF of a lopsided neckpinch geometry through the Type-1 singularity using surgery and yielding the geometry as a direct product of two 3-spheres. We use axial symmetry of our model to suppress one dimension and the resulting two-lobed geometry can be visualized in Euclidean 3-space (our evolution was fortunately isometrically embeddable in R 3 ). The middle 3’rd and 4’th figure occur at the same time (t = 183.0) in the evolution. They illustrate the explicit manifold surgery, where the spherical caps (two icosahedrons )are placed on the ends of the left and right lobes. This is the first numerical illustration of Thurston’s geometrization procedure that we are aware of. This surface has 3438 edges, 1580 triangle-based frustum blocks and 960 vertices, although symmetry reduces the number of edges to 80 icosahedral {si} edges and 79 axial {ai} edges.

Paul M. Alsing1
, Warner A. Miller2† & Shing-Tung Yau3
1 Air Force Research Laboratory, Information Directorate, Rome, NY 13441
2 Department of Physics, Florida Atlantic University, Boca Raton, FL 33431
3 Department of Mathematics, Harvard University, Cambridge, MA 02138

Hamilton의 Ricci 흐름 (RF) 방정식은 최근 d 차원 부분 선형 (PL) 단순 기하학의 모서리 길이에 대한 자율 1 차 비선형 미분 방정식의 희소 결합 시스템으로 표현되었습니다. 더 최근에, 이 DRF (Discrete Ricci Flow) 방정식 시스템은 각 에지에 연결된 Forman-Ricci 텐서를 명시적으로 구성하여 1 차 미분 연산자를 대각화하고 각 시간 단계에서 큰 희소 행렬을 반전할 필요가 없도록 더욱 단순화되었습니다.

우리는 최근에 이러한 방정식이 축 대칭 3-기하 구조에 대해 해당 연속체 RF 방정식에 수렴한다는 것을 분석적으로나 수치적으로 보여주었습니다.

우리는 여기서 이러한 DRF 방정식이 수술을 사용하여 Type-1 넥 핀치 특이점을 통해 명시적으로 통합함으로써 이산 3D 축 대칭 넥 핀치 형상에 대한 Thurston의 형상화 절차에 대한 명시적인 수치 실현을 보여줍니다.

진화를 완료하려면 입방 스플라인 기반 적응형 메시가 필요했습니다. 우리의 수치적으로 효율적인 시뮬레이션은 충분히 조여진 축 대칭 형상의 예상되는 Thurston 분해를 고유 한 기하학적 구조 (각각 3 구 형상으로 축소되는 두 로브의 직접 곱)로 산출합니다. 곡률의 구조는 Forman이 그래프에서 Ricci 곡률을 표현할 때 나타나는 정점 및 가장자리 가중치 중 하나를 더 잘 알리는 데 사용될 수 있습니다.

Thurstons Geometrization의 실현 : Surgery을 통한 Discrete Ricci Flow

FIG. 2: A two dimensional representation of the 3D neckpinch geometry of Angenent and Knopf (continuum on top, and discrete on bottom). In 3D the continuum cross-sections are 3-spheres and not circles, and in our discrete model the cross sections are icosahedrons and not hexagons. The 3D cells are triangle-based frustum blocks as opposed to the trapezoids depicted in the bottom of the figure. Here the variable ac measures the proper distance from the equator, and s is the length of the icosahedron edges
FIG. 2: A two dimensional representation of the 3D neckpinch geometry of Angenent and Knopf (continuum on top, and discrete on bottom). In 3D the continuum cross-sections are 3-spheres and not circles, and in our discrete model the cross sections are icosahedrons and not hexagons. The 3D cells are triangle-based frustum blocks as opposed to the trapezoids depicted in the bottom of the figure. Here the variable ac measures the proper distance from the equator, and s is the length of the icosahedron edges
FIG. 3: An illustration of the icosahedron neckpinch geometry for nine cross-sectional icosahedra (top), and its dual dodecahedral lattice (bottom). The lattice is composed of triangle-based frustum blocks, and the dual lattice is composed of pentagonal-based frustum blocks. The expressions for the sectional, scalar, and Ricci curvature uses the dual lattice with its dodecahedral cross sections.
FIG. 3: An illustration of the icosahedron neckpinch geometry for nine cross-sectional icosahedra (top), and its dual dodecahedral lattice (bottom). The lattice is composed of triangle-based frustum blocks, and the dual lattice is composed of pentagonal-based frustum blocks. The expressions for the sectional, scalar, and Ricci curvature uses the dual lattice with its dodecahedral cross sections.
FIG. 4: The RF of a lopsided neckpinch geometry through the Type-1 singularity using surgery and yielding the geometry as a direct product of two 3-spheres. We use axial symmetry of our model to suppress one dimension and the resulting two-lobed geometry can be visualized in Euclidean 3-space (our evolution was fortunately isometrically embeddable in R 3 ). The middle 3’rd and 4’th figure occur at the same time (t = 183.0) in the evolution. They illustrate the explicit manifold surgery, where the spherical caps (two icosahedrons )are placed on the ends of the left and right lobes. This is the first numerical illustration of Thurston’s geometrization procedure that we are aware of. This surface has 3438 edges, 1580 triangle-based frustum blocks and 960 vertices, although symmetry reduces the number of edges to 80 icosahedral {si} edges and 79 axial {ai} edges.
FIG. 4: The RF of a lopsided neckpinch geometry through the Type-1 singularity using surgery and yielding the geometry as a direct product of two 3-spheres. We use axial symmetry of our model to suppress one dimension and the resulting two-lobed geometry can be visualized in Euclidean 3-space (our evolution was fortunately isometrically embeddable in R 3 ). The middle 3’rd and 4’th figure occur at the same time (t = 183.0) in the evolution. They illustrate the explicit manifold surgery, where the spherical caps (two icosahedrons )are placed on the ends of the left and right lobes. This is the first numerical illustration of Thurston’s geometrization procedure that we are aware of. This surface has 3438 edges, 1580 triangle-based frustum blocks and 960 vertices, although symmetry reduces the number of edges to 80 icosahedral {si} edges and 79 axial {ai} edges.
FIG. 5: A 2-dimensional cross section of a lopsided neckpinch geometry evolving under RF through the Type-1 singularity. Surgery yields two disconnected 3D ovoids and each becomes spherical under the RF evolution. The resulting geometry is a direct product of two 3-spheres. As the lobed geometry collapses a pinch occurs at t= 183. At this point we remove the axial edges at the pinch and cap each end of the left and right lobe with a new icosahedra. These two surfaces (pre and post surgery) are the 3rd and 4th layers inside the initial surface. After surgery, we remesh both the left and right 3-dimensional ovoids using cubic spline interpolation. This is, to our knowledge, the first numerical realization for PL manifolds of Thurston’s geometrization procedure. This particular surface has 3348 edges, 1580 triangle-based frustum blocks and 960 vertices, although symmetry reduces the number of edges to 80 icosahedral {si} edges and 79 axial {ai} edges.
FIG. 5: A 2-dimensional cross section of a lopsided neckpinch geometry evolving under RF through the Type-1 singularity. Surgery yields two disconnected 3D ovoids and each becomes spherical under the RF evolution. The resulting geometry is a direct product of two 3-spheres. As the lobed geometry collapses a pinch occurs at t= 183. At this point we remove the axial edges at the pinch and cap each end of the left and right lobe with a new icosahedra. These two surfaces (pre and post surgery) are the 3rd and 4th layers inside the initial surface. After surgery, we remesh both the left and right 3-dimensional ovoids using cubic spline interpolation. This is, to our knowledge, the first numerical realization for PL manifolds of Thurston’s geometrization procedure. This particular surface has 3348 edges, 1580 triangle-based frustum blocks and 960 vertices, although symmetry reduces the number of edges to 80 icosahedral {si} edges and 79 axial {ai} edges.
FIG. 6: After the manifold surgery the lobe was closed using a spherical cap with proper matching conditions as illustrated in this figure. This involved reassigning the values to two of the s variables and two of the a values. This procedure offers no essential advantage over the simpler procedure consisting of just capping the surgery with an icosahedron and remeshing.
FIG. 6: After the manifold surgery the lobe was closed using a spherical cap with proper matching conditions as illustrated in this figure. This involved reassigning the values to two of the s variables and two of the a values. This procedure offers no essential advantage over the simpler procedure consisting of just capping the surgery with an icosahedron and remeshing.
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