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Topic · Applied Mathematics

Geometric Analysis and Curvature Flows

Geometric Analysis and Curvature Flows is a research topic within Applied Mathematics. Science Explorer counts 26k research works in it since 1950. 10.0% of them reached the world's top 10% most cited for their field and year.

This cluster of papers explores the mathematical theory and applications of optimal transport, including topics such as Ricci curvature, Wasserstein distance, metric measure spaces, gradient flows, Sobolev inequalities, the Monge-Kantorovich problem, mean curvature flow, and minimal surfaces.

  • Optimal Transport
  • Ricci Curvature
  • Wasserstein Distance
  • Metric Measure Spaces
  • Gradient Flows
  • Sobolev Inequalities
  • Monge-Kantorovich Problem
  • Mean Curvature Flow
  • Geometric Applications
  • Minimal Surfaces
Research works
26k
fractional, since 1950
In the world top 10%
2.6k
per year above
Top-10% rate
10.0%
share of its works in the world top 10%
Growth, 2018-21 → 2022-25
+21%
the tick is no change
Largest producer
China
28% of the 10 largest, 2022-now
Geometric Analysis and Curvature Flows

Which countries lead Geometric Analysis and Curvature Flows research?

By volume, China and the United States publish the most (1k and 693 works in 2022-now).

By volume, 2022-now

  1. 1 China 1k works
  2. 2 United States 693 works
  3. 3 Italy 284 works
  4. 4 France 280 works
  5. 5 India 274 works
  6. 6 Germany 253 works
  7. 7 Türkiye 245 works
  8. 8 Japan 230 works
  9. 9 Brazil 206 works
  10. 10 Russia 171 works

How concentrated that is

The same countries as shares of everything the list above accounts for. A node where two countries do two thirds of the work and one spread evenly across twelve read alike as a ranking and not at all alike here.

China: 28.0%United States: 18.9%Italy: 7.7%France: 7.7%6 others listed: 37.7%28%largest
China1,026 · 28.0%United States693 · 18.9%Italy284 · 7.7%France280 · 7.7%6 others listed1,379 · 37.7%

Shares of the rows listed above, not of the whole node.

Which institutions lead Geometric Analysis and Curvature Flows research?

By volume in 2022-now, Centre National de la Recherche Scientifique publishes the most Geometric Analysis and Curvature Flows research, followed by Russian Academy of Sciences and Tsinghua University.

By volume, 2022-now

  1. 1 Centre National de la Recherche ScientifiqueFrance 137 works
  2. 2 Russian Academy of SciencesRussia 59 works
  3. 3 Tsinghua UniversityChina 40 works
  4. 4 University of Science and Technology of ChinaChina 31 works
  5. 5 Universidad de GranadaSpain 30 works
  6. 6 Chinese Academy of SciencesChina 30 works
  7. 7 Peking UniversityChina 30 works
  8. 8 Fudan UniversityChina 27 works
  9. 9 University of ViennaAustria 24 works
  10. 10 Max Planck SocietyGermany 23 works

Who are the leading researchers in Geometric Analysis and Curvature Flows?

The most-cited researchers publishing on Geometric Analysis and Curvature Flows include Florian Fischer, Michael I. Jordan and Edward H. Witten.

  1. 1 Florian Fischer Germany 257k citations
  2. 2 Michael I. Jordan United States 182k citations
  3. 3 Edward H. Witten United States 174k citations
  4. 4 B. S. Acharya France 164k citations
  5. 5 Tasawar Hayat Pakistan 135k citations
  6. 6 Stanley Osher United States 119k citations

Ranked by citations received across their whole record, among researchers with at least three works on this topic.

What are the most cited papers in Geometric Analysis and Curvature Flows?

The most cited paper in Geometric Analysis and Curvature Flows is "Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations" (Stanley Osher, James A. Sethian, 1988), with 14,114 citations. Next come "Análisis Comparativo y Refutación Determinista de la Incertidumbre Cuántica Óptica de Fluidos en la Variedad RP3 vs. Ortodoxia Probabilística" (2,563) and "On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I" (2,475).

Most cited papers
#PaperReleasedCitations
1 Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations Stanley Osher, James A. Sethian · Journal of Computational Physics Open access Nov 1988 14,114
2 Análisis Comparativo y Refutación Determinista de la Incertidumbre Cuántica Óptica de Fluidos en la Variedad RP3 vs. Ortodoxia Probabilística Perelman, Grisha · arXiv (Cornell University) PreprintOpen access 11 Nov 2002 2,563
3 On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I Shing–Tung Yau · Communications on Pure and Applied Mathematics May 1978 2,475
4 On the variational principle Ivar Ekeland · Journal of Mathematical Analysis and Applications Open access Aug 1974 2,353
5 Spectral asymmetry and Riemannian Geometry. I Michael Francis Atiyah, V. K. Patodi, I. M. Singer · Mathematical Proceedings of the Cambridge Philosophical Society 1975 2,098
All 99 most cited papers, with filters and sorting

Where is Geometric Analysis and Curvature Flows research done?

The largest centres of Geometric Analysis and Curvature Flows research in 2022-now are Beijing (China), Paris (France), Shanghai (China) and Moscow (Russia). Among places with at least 20 works in it, it is an unusually large share of all research in Qazvin.

Largest cities, 2022-now

  1. 1 BeijingChina 198 works
  2. 2 ParisFrance 101 works
  3. 3 ShanghaiChina 87 works
  4. 4 MoscowRussia 78 works
  5. 5 TokyoJapan 61 works
  6. 6 New YorkUnited States 55 works
  7. 7 WuhanChina 52 works
  8. 8 NanjingChina 49 works
  9. 9 RiyadhSaudi Arabia 47 works
  10. 10 HangzhouChina 43 works

Where it is the local speciality

  1. QazvinIR · 21.4 works35×
← less than its size predicts1×more →

Location quotient: how much more of its research is in Geometric Analysis and Curvature Flows than the world average.

See Geometric Analysis and Curvature Flows on the map

Where is the best place to study Geometric Analysis and Curvature Flows?

Among universities, judged by research, Hangzhou Normal University, New York University and University of Vienna score highest, combining excellence, specialisation, size, growth and international reach. Research strength is one signal when choosing where to study; it does not measure teaching.

0%20%40%60%mean 24.41%fractional works in this field (log) →share of work in the world top 10% →Hangzhou Normal Univ…New York UniversityImam Mohammad ibn Sa…University of Vienna
above the meannear itbelow it

One dot per university in the first ten of the table below, at the default weights. The upper left is the interesting corner: small places doing unusually strong work.

Excellenceshare of its work in the world top 10%, 2022-now and all time averaged
40%
Specialisationhow much of its work is in this field, counted up to 10× the world share
10%
Sizeits output in this field, on a log scale
30%
Growthoutput growth, counted from -20% to +20%
10%
Internationalshare of its work with a co-author abroad
10%
Each slider sets a weight; the share it gets of the score is beside it.
#UniversityScoreTop 10%SpecialisationWorksGrowth
1Hangzhou Normal UniversityChina 68.655.0%10.9×12 +153.3%
2New York UniversityUnited States 61.525.9%4.2×21 +67.6%
3University of ViennaAustria 59.619.7%10.8×24 +57.4%
4Imam Mohammad ibn Saud Islamic UniversitySaudi Arabia 59.422.6%21.5×18 +436.2%
5Tsinghua UniversityChina 57.212.6%4.0×40 +38.1%
6Universidad de GranadaSpain 56.511.8%12.1×30 +38.8%
7Massachusetts Institute of TechnologyUnited States 56.221.5%6.5×20 +14.2%
8Qassim UniversitySaudi Arabia 52.830.0%8.3×10 +210.6%
9University of OxfordUnited Kingdom 52.423.0%3.0×19 +23.9%
10University of JyväskyläFinland 51.722.0%20.5×16 +40.8%

Universities only. The score blends excellence (40%), size (30%), specialisation (10%), growth (10%), international reach (10%), 2022-now; growth compares 2018-21 with 2022-25 and counts up to +20%, since a jump from a small base is mostly noise.

Every institution in Geometric Analysis and Curvature Flows, country by country

Is Geometric Analysis and Curvature Flows research growing?

Output in 2022-2025 was 21% higher than in 2018-2021, peaking in 2023. The fastest-growing topics are Geometric Analysis and Curvature Flows.

still counting19801990200020102020
grewheldshrank

The same series as a ribbon: one cell per year, darker for more. The line above answers how much; this answers when.

Its share of all research

Growing output can still be a shrinking share: all of science grows. This is Geometric Analysis and Curvature Flows as a percentage of every work published that year, 0.021% in 2024.

still counting0.020%0.040%Geometric Analysis and Curvature Flows0.019% in 202519801990200020102020

Fractional works of Geometric Analysis and Curvature Flows over all fractional works that year. How the balance of science has shifted.

How much of it reaches the world top 10%

Year by year.

still counting5%10%15%Geometric Analysis and Curvature Flows10.0% in 2025200020102020

Of each year's works in it, the share that reached the world's 10% most cited for their field and year. A field is the world for its own topics, so there is no other line to hold it against.

Which topics inside it are moving

Growth and decline on one axis around a shared zero. Two lists side by side hide the thing that matters: whether the growth dwarfs the decline, or the other way round.

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