弗蘭克·摩根（Frank Morgan）是一位享譽(yù)國(guó)際的美國(guó)數(shù)學(xué)家，以其深厚的學(xué)術(shù)造詣和突破性的研究成果在數(shù)學(xué)界享有盛譽(yù)。摩根教授在哈佛大學(xué)完成了他的學(xué)士學(xué)位學(xué)習(xí)，隨后在普林斯頓大學(xué)攻讀碩士和博士學(xué)位，師從幾何分析學(xué)家威廉·特勞布里奇（William H. Trotter）教授。目前，摩根教授擔(dān)任美國(guó)威廉斯學(xué)院的數(shù)學(xué)教授，同時(shí)，他還曾在多所知名大學(xué)擔(dān)任訪問教授和客座教授，積累了豐富的數(shù)學(xué)教學(xué)和研究經(jīng)驗(yàn)。他的研究興趣廣泛，聚焦于曲線和曲面的幾何性質(zhì)、測(cè)度論在高維空間中的創(chuàng)新應(yīng)用，以及微分幾何中的復(fù)雜變分問題。摩根教授的研究不僅深入理論層面，還積極探索其在多個(gè)領(lǐng)域的實(shí)際應(yīng)用，展現(xiàn)了其跨學(xué)科的綜合能力。
圖書目錄
Contents Preface vii Part I: Basic Theory 1 1Geometric Measure Theory 3 2Measures 11 3Lipschitz Functions and Recti?able Sets 25 4Normal and Recti?able Currents 39 5The Compactness Theorem and the Existence of Area-Minimizing Surfaces 61 6Examples of Area-Minimizing Surfaces 69 7The Approximation Theorem 79 8Survey of Regularity Results 83 9Monotonicity and Oriented Tangent Cones 89 10The Regularity of Area-Minimizing Hypersurfaces 97 11Flat Chains Moduloν, Varifolds, and -Minimal Sets 105 12Miscellaneous Useful Results 111
Part II: Applications 119 13Soap Bubble Clusters 121 14Proof of Double Bubble Conjecture 143 15The Hexagonal Honeycomb and Kelvin Conjectures 159 16Immiscible Fluids and Crystals 173 17Isoperimetric Theorems in General Codimension 179 18Manifolds with Density and Perelman’s Proof of the Poincaré Conjecture 183 19Double Bubbles in Spheres, Gauss Space, and Tori 197 20The Log-Convex Density Theorem 205
Solutions to Exercises 213 Bibliography 235 Index of Symbols 255 Name Index 257 Subject Index 259
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